Quaternion Algebraic Geometry

Dominic Widdows

June 5, 2006 Abstract

QUATERNION ALGEBRAIC GEOMETRY

DOMINIC WIDDOWS

St Anne’s College, Oxford

Thesis submitted Hilary Term, 2000, in support of application to supplicate for the degree of D.Phil.

This thesis is a collection of results about hypercomplex and quaternionic manifolds, focussing on two main areas. These are exterior forms and double complexes, and the ‘algebraic geometry’ of hypercomplex manifolds. The latter area is strongly influenced by techniques from quaternionic algebra. A new double complex on quaternionic manifolds is presented, a quaternionic version of the Dolbeault complex on a complex manifold. It arises from the decomposition of real-valued exterior forms on a quaternionic manifold M into irreducible representations of Sp(1). This decomposition gives a double complex of differential forms and operators ∼ as a result of the Clebsch-Gordon formula Vr⊗V1 = Vr+1⊕Vr−1 for Sp(1)-representations. The properties of the double complex are investigated, and it is established that it is elliptic in most places. Joyce has created a new theory of quaternionic algebra [J1] by defining a quaternionic tensor product for AH-modules (H-modules equipped with a special real subspace). The 1 theory can be described using sheaves over CP , an interpretation due to Quillen [Q]. AH-modules and their quaternionic tensor products are classified. Stable AH-modules are described using Sp(1)-representations. This theory is especially useful for describing hypercomplex manifolds and forming close analogies with complex geometry. Joyce has defined and investigated q-holomorphic functions on hypercomplex manifolds. There is also a q-holomorphic cotangent space which again arises as a result of the Clebsch-Gordon formula. AH-module bundles are defined and their q-holomorphic sections explored. Quaternion-valued differential forms on hypercomplex manifolds are of special interest. Their decomposition is finer than that of real forms, giving a second double complex with special advantages. The cohomology of these complexes leads to new invariants of compact quaternionic and hypercomplex manifolds. Quaternion-valued vector fields are also studied, and lead to the definition of quaternionic Lie algebras. The investigation of finite-dimensional quaternionic Lie algebras allows the calculation of some simple quaternionic cohomology groups. Acknowledgements

I would like to express gratitude, appreciation and respect for my supervisor Do- minic Joyce. Without his inspiration as a mathematician this thesis would not have been conceived; without his patience and friendship it would certainly not have been completed.

Dedication

To the memory of Dr Peter Rowe, 1938-1998, who taught me relativity as an undergraduate in Durham. His determination to teach concepts before examination techniques introduced me to diﬀerential geometry.

i Contents

Introduction 1

1 The Quaternions and the Group Sp(1) 4 1.1 The Quaternions ...... 4 1.2 The Lie Group Sp(1) and its Representations ...... 9 1.3 Diﬃculties with the Quaternions ...... 12

2 Quaternionic Differential Geometry 17 2.1 Complex, Hypercomplex and Quaternionic Manifolds ...... 17 2.2 Differential Forms on Complex Manifolds ...... 20 2.3 Differential Forms on Quaternionic Manifolds ...... 23

3 A Double Complex on Quaternionic Manifolds 25 3.1 Real forms on Complex Manifolds ...... 25 3.2 Construction of the Double Complex ...... 28 3.3 Ellipticity and the Double Complex ...... 32 3.4 Quaternion-valued forms on Hypercomplex Manifolds ...... 40

4 Developments in Quaternionic Algebra 42 4.1 The Quaternionic Algebra of Joyce ...... 42 4.2 Duality in Quaternionic Algebra ...... 48 4.3 Real Subspaces of Complex Vector Spaces ...... 52 4.4 K-modules ...... 53 4.5 The Sheaf-Theoretic approach of Quillen ...... 56

5 Quaternionic Algebra and Sp(1)-representations 61 5.1 Stable AH-modules and Sp(1)-representations ...... 61 5.2 Sp(1)-Representations and the Quaternionic Tensor Product ...... 68 5.3 Semistable AH-modules and Sp(1)-representations ...... 74 5.4 Examples and Summary of AH-modules ...... 77 6 Hypercomplex Manifolds 80 6.1 Q-holomorphic Functions and H-algebras ...... 81 6.2 The Quaternionic Cotangent Space ...... 86 6.3 AH-bundles ...... 90 6.4 Q-holomorphic AH-bundles ...... 93

ii 7 Quaternion Valued Forms and Vector Fields 100 7.1 The Quaternion-valued Double Complex ...... 101 7.2 Vector Fields and Quaternionic Lie Algebras ...... 109 7.3 Hypercomplex Lie groups ...... 112

References 117

iii Introduction

This aim of this thesis is to describe and develop various aspects of quaternionic algebra and geometry. The approach is based upon two pillars, namely the differential geometry of quaternionic manifolds and Joyce’s recent theory of quaternionic algebra. Contribu- tions are made to both fields of study, enabling these strands to be woven together in describing the algebraic geometry of hypercomplex manifolds. A recurrent theme throughout will be representations of the group Sp(1) of unit quaternions. Our contribution to the theory of quaternionic manifolds relies on decomposing the Sp(1)-action on exterior forms, whilst the main new insight in quaternionic algebra is that the most important building blocks of Joyce’s theory are best described and manipulated as Sp(1)-representations. The importance of Sp(1)-representations to both areas is chiefly responsible for the successful synthesis of methods in the work on hypercomplex manifolds. Another frequent source of motivation is the behaviour of the complex numbers. Many situations in complex algebra and geometry have interesting quaternionic analogues. Aspects of complex geometry can often be described using the group U(1) of unit complex numbers; replacing this with the group Sp(1) can lead directly to quaternionic versions. The decomposition of exterior forms on quaternionic manifolds is precisely such an example, as is all the work on q-holomorphic functions and forms on hypercomplex manifolds. On the other hand, Joyce’s quaternionic algebra is such a rich theory precisely because real subspaces of quaternionic vector spaces behave so differently from real subspaces of complex vector spaces. Much of the original work presented is enticingly simple — indeed, I have often felt both surprised and privileged that it has not been carried out before. One of the main explanations for this is the relative unpopularity suffered by the quaternions in the 20th century. This situation has left various aspects of quaternionic behaviour unexplored. To help understand the reasons for this omission and the consequent opportunities for development, the first chapter is devoted to a survey of the history of the quaternions and their applications. Background material also includes an introduction to the group Sp(1) and its representations. The irreducible representations on complex vector spaces and their tensor products are described, as are real and quaternionic representations. Chapter 2 is about quaternionic structures in differential geometry. The approach is based on the work of Salamon [S3]. Taking complex manifolds as a model, hypercomplex manifolds (those possessing a torsion-free GL(n, H)-structure) are defined, followed by the broader class of quaternionic manifolds (those possessing a torsion-free Sp(1)GL(n, H)-structure). After reviewing the Dolbeault complex, we consider the decomposition of differential forms on quaternionic manifolds, including an important elliptic complex discovered by Salamon upon which the integrability of the quaternionic

1 structure depends. This complex is in fact the top row of a hitherto undiscovered double complex on quaternionic manifolds, which is the subject of Chapter 3. As an Sp(1)-representation, 4n ∗ ∼ the cotangent space of a quaternionic manifold M takes the form T M = 2nV1, where 2 V1 is the basic representation of Sp(1) on C . Decomposition of the induced Sp(1)- representation on ΛkT ∗M is a simple process achieved by considering weights. That this decomposition gives rise to a double complex results from the Clebsch-Gordon formula ∗ ∼ ∼ Vr ⊗T M = Vr ⊗2nV1 = 2n(Vr+1 ⊕Vr−1). The new double complex is shown to be elliptic everywhere except along its bottom row, consisting of the basic representations V1 and the trivial representations V0. This double complex presents us with new quaternionic cohomology groups. In the fourth chapter (which is partly a summary of the work of Joyce [J1] and Quillen [Q]) we move to our other major area of interest, the theory of quaternionic algebra. The building blocks of this theory are H-modules equipped with a special real subspace. Such an object is called an AH-module. Joyce has discovered a canonical tensor product operation for AH-modules which is both associative and commutative. Using ideas from Quillen’s work, we classify AH-modules up to isomorphism. Dual AH-modules are defined and shown to have interesting properties. Particularly well- behaved is the category of stable AH-modules. In Chapter 5 it is shown that all stable AH-modules and their duals are conveniently described using Sp(1)-representations. The resulting theory is ideally adapted for describing hypercomplex geometry, a process begun in Chapter 6. A hypercomplex manifold M has a triple of global anticom- muting complex structures which can be identified with the imaginary quaternions. This identification enables tensors on hypercomplex manifolds to be treated using the techniques of quaternionic algebra. Joyce has already used such an approach to define and investigate q-holomorphic functions on hypercomplex manifolds, which are seen as the quaternionic analogue of holomorphic functions. There is a natural product map on the AH-module of q-holomorphic functions, which gives the q-holomorphic functions an algebraic structure which Joyce calls an H-algebra. Using the Sp(1)-version of quaternionic algebra, we define a natural splitting of the ∗ ∼ quaternionic cotangent space H ⊗ T M = A ⊕ B, and show that q-holomorphic functions are precisely those whose differentials take values in A ⊂ H ⊗ T ∗M. The bundle A is hence defined to be the q-holomorphic cotangent space of M. These spaces are examples of AH-module bundles or AH-bundles, which we discuss. Several parallels with complex geometry arise. There are q-holomorphic AH-bundles with q-holomorphic sections. Q-holomorphic sections are described using the quaternionic tensor product and the q-holomorphic cotangent space, and seen to form an H-algebra module over the q-holomorphic functions. In the final chapter, such methods are applied to quaternion-valued tensors on hypercomplex manifolds. The double complex of Chapter 3 is revisited and adapted to quaternion-valued differential forms. The global complex structures give an extra decom- ∗ ∼ position which generalises the splitting H ⊗ T M = A ⊕ B, further refining the double complex. The quaternion-valued double complex has advantages over the real-valued version, being elliptic in more places. The top row of the quaternion-valued double complex is particularly well-adapted to quaternionic algebra, which presents close parallels with the Dolbeault complex and motivates the definition of q-holomorphic k-forms.

2 Quaternion-valued vector fields are also interesting. The quaternionic tangent space ∼ splits as H ⊗ TM = Ab⊕ Bb in the same way as the cotangent space. Vector fields taking values in Ab are closed under the quaternionic tensor product and Lie bracket, a result which depends upon the integrability of the hypercomplex structure. This is the quaternionic analogue of the statement that on a complex manifold, the (1, 0) vector fields are closed under the Lie bracket. The vector fields in question therefore form a quaternionic Lie algebra, a new concept which we introduce. Interesting finite-dimensional quaternionic Lie algebras are used to calculate some quaternionic cohomology groups on Lie groups with left-invariant hypercomplex structures.

3 Chapter 1

The Quaternions and the Group Sp(1)

1.1 The Quaternions

The quaternions H are a four-dimensional real algebra generated by the identity element 1 and the symbols i1, i2 and i3, so H = {r0 +r1i1 +r2i2 +r3i3 : r0, . . . , r3 ∈ R}. Quaternions are added together component by component, and quaternion multiplication is given by the quaternion relations

2 2 2 i1i2 = −i2i1 = i3, i2i3 = −i3i2 = i1, i3i1 = −i1i3 = i2, i1 = i2 = i3 = −1 (1.1) and the distributive law. The quaternion algebra is not commutative, though it does obey the associative law. The quaternions are a division algebra (an algebra with the property that ab = 0 implies that a = 0 or b = 0 ).

• Deﬁne the imaginary quaternions I = hi1, i2, i3i. The symbol I is not standard, but we will use it throughout.

• Deﬁne the conjugateq ¯ of q = q0 + q1i1 + q2i2 + q3i3 byq ¯ = q0 − q1i1 − q2i2 − q3i3. Then (pq) =q ¯p¯ for all p, q ∈ H.

• Deﬁne the real and imaginary parts of q by Re(q) = q0 ∈ R and Im(q) = q1i1 + q2i2 + q3i3 ∈ I. As with complex numbers,q ¯ = Re(q) − Im(q).

• We regard the real numbers R as a subﬁeld of H, and the quaternions as a direct ∼ sum H = R ⊕ I.

2 2 2 2 • Let q = q1i1 + q2i2 + q3i3 ∈ I. Then q = −1 if and only if q1 + q2 + q3 = 1, so the set of ‘quaternionic square-roots of minus-one’ is naturally isomorphic to the 2-sphere S2. We shall often identify these sets, writing ‘q ∈ S2’ as a shorthand for ‘q ∈ H : q2 = −1’.

• If q ∈ S2 then h1, qi is a subﬁeld of H isomorphic to C. We shall call this subﬁeld Cq.

4 1.1.1 A History of the Quaternions The quaternions were discovered by the Irish mathematician and physicist, William Rowan Hamilton (1805-1865), 1 whose contributions to mechanics are well-known and widely used. By 1835 Hamilton had helped to win acceptance for the system of complex numbers by showing that calculations with complex numbers are equivalent to calculations with ordered pairs of real numbers, governed by certain rules. At the time, complex 2 numbers were being applied very eﬀectively to problems in the plane R . To Hamilton, the next logical step was to seek a similar 3-dimensional number system which would 3 revolutionise calculations in R . For years, he struggled with this problem. In a touching letter to his son [H1], 2 dated shortly before his death in 1865, Hamilton writes:

Every morning, on my coming down to breakfast, your brother and yourself used to ask me: “Well, Papa, can you multiply triplets?” Whereto I was always obliged to reply, with a sad shake of the head, “No, I can only add and subtract them”.

For several years, Hamilton tried to manipulate the three symbols 1, i and j into an algebra. He ﬁnally realised that the secret was to introduce a fourth dimension. On 16th October, 1843, whilst walking with his wife, he had a ﬂash of inspiration. In the same letter, he writes:

An electric circuit seemed to close, and a spark ﬂashed forth, the herald (as I foresaw immediately) of many long years to come of deﬁnitely directed thought and work ... I pulled out on the spot a pocket-book, which still exists, and made an entry there and then. Nor could I resist the impulse — unphilosophical as it may have been — to cut with a knife on the stone of Brougham Bridge, as we passed it, the fundamental formula with the symbols i, j, k: i2 = j2 = k2 = ijk = −1, which contains the solution of the problem, but of course, as an inscription, has long since mouldered away.

Substituting i1, i2, i3 for i, j, k, this gives the quaternion relations (1.1). Rarely do we possess such a clear account of the genesis of a piece of mathematics. Most mathematical theories are invented gradually, and only after years of development can they be presented in a lecture course as a deﬁnitive set of axioms and results. The quaternions, on the other hand, “started into life, or light, full grown, on the 16th of Ocober, 1843...” 3 “Less than an hour elapsed” before Hamilton obtained leave of the Council of the Royal Irish Academy to read a paper on quaternions. The next day, Hamilton wrote a detailed letter to his friend and fellow mathematician John T. Graves 1Letters suggest that both Euler and Gauss were aware of the quaternion relations (1.1), though neither of them published the disovery [EKR, p. 192]. 2Copies of Hamilton’s most signiﬁcant letters and papers concerning quaternions are currently avail- able on the internet at www.maths.tcd.ie/pub/HistMath/People/Hamilton 3Letter to Professor P.G.Tait, an excerpt of which can be found on the same website as [H1].

5 [H2], giving us a clear account of the train of research which led him to his breakthrough. The discovery was published within a month on the 13th of November [H3]. The timing of the discovery amplified its impact upon Hamilton and his followers. The only other algebras known in 1843 were the real and complex numbers, both of which can be regarded as subalgebras of the quaternions. (It was not until 1858 that Cayley introduced matrices, and showed that the quaternion algebra could be realised as a subalgebra of the complex-valued 2 × 2 matrices.) As a result, Hamilton became the figurehead of a school of ‘quaternionists’, whose fervour for the new numbers far exceeded their usefulness. Hamilton believed his discovery to be of similar importance to that of the infinitesimal calculus, and devoted the rest of his career exclusively to its study. Echos of this zeal could still be heard this century; for example, while Eamon de Valera was President of Ireland (from 1959 to 1973), he would attend mathematical meetings whenever their title contained the word ‘quaternions’! Such excesses were bound to provoke a reaction, especially as it became clear that the quaternions are just one example of a number of possible algebras. Lord Kelvin, the famous Scottish physicist, once remarked that “Quaternions came from Hamilton after all his really good work had been done; and though beautifully ingenious, have been an unmixed evil to those who have touched them in any way” [EKR, p.193]. A belief that quaternions are somehow obsolete is often tacitly accepted to this day. This is far from the case. The quaternions remain the simplest algebra after the real and complex numbers. Indeed, the real numbers R, the complex numbers C and the quaternions H are the only associative division algebras, as was proved by Georg Frobenius in 1878: and amongst these the quaternions are the most general. The discovery of the quaternions provided enormous stimulation to algebraic research and it is thought that the term ‘associative’ was coined by Hamilton himself [H3, p.5] to describe quaternionic behaviour. Investigation into the nature of and constraints imposed by algebraic properties such as associativity and commutativity was greatly accelerated by the discovery of the quaternions. The quaternions themselves have been used in various areas of mathematics. Most recently, quaternions have enjoyed prominence in computer science, because they are the simplest algebraic tool for describing rotations in three and four dimensions. Certainly, the numbers have fallen short of the early expectations of the quaternionists. However, quaternions do shine a light on certain areas of mathematics, and those who become familiar with them soon come to appreciate an intricacy and beauty which is all their own.

1.1.2 Quaternions and Matrices

In this section we will make use of the older notation i = i1, j = i2, k = i3. This makes it easy to interpret i as the standard complex ‘square root of −1’ and j as a ‘structure 2 map’ on the complex vector space C . It is well-known that the quaternions can be written as real or complex matrices, because there are isomorphisms from H into subalgebras of Mat(4, R) and Mat(2, C). The former of these is given by the mapping

6   q0 q1 q2 q3  −q1 q0 −q3 q2  q0 + q1i + q2j + q3k 7→   .  −q2 q3 q0 −q1  −q3 −q2 q1 q0

More commonly used is the mapping into Mat(2, C). We can write every quaternion as a pair of complex numbers, using the equation

q0 + q1i + q2j + q3k = (q0 + q1i) + (q2 + q3i)j. (1.2)

∼ 2 In this way we obtain the expression q = α+βj ∈ H = C . The map j : α+βj 7→ −β¯+αj ¯ 2 2 ∼ 2 is a conjugate-linear involution of C with j = −1. This identiﬁcation H = C is not uniquely determined: each q ∈ S2 determines a similar isomorphism. Having written this down, it is easy to form the map

α β ι : → H ⊂ Mat(2, ) α + βj 7→ . (1.3) H C −β¯ α¯

The quaternion algebra can thus be realised as a real subalgebra of Mat(2, C), using the identiﬁcations

1 0 i 0 0 1 0 i 1 = i = i = i = . (1.4) 0 1 1 0 −i 2 −1 0 3 i 0

Note that the squared norm qq¯ of a quaternion q is the same as the determinant of the matrix ι(q) ∈ H. The isomorphism ι gives an easy way to deduce that H is an associative division algebra; the inverse of any nonzero matrix A ∈ H is also in H, and the only matrix in H whose determinant is zero is the zero matrix.

1.1.3 Simple Applications of the Quaternions There are a number of ways in which quaternions can be used to express mathematical ideas. In many cases, a quaternionic description preﬁgures more modern descriptions. We will outline two main areas – vector analysis and Euclidean geometry. An excellent and readable account of most of the following can be found in Chapter 7 of [EKR]. Every quaternion can be uniquely written as the sum of its real and imaginary parts. 3 If we identify the imaginary quaternions I with the real vector space R , we can consider each quaternion q = q0+q1i1+q2i2+q3i3 as the sum of a scalar part q0 and a vectorial part 3 (q1, q2, q3) ∈ R (indeed, it is in this context that the term ‘vector’ ﬁrst appears [H4]). If we multiply together two imaginary quaternions p, q ∈ I, we obtain a quaternionic 3 version of the scalar product and vector product on R , as follows: ∼ pq = −p · q + p ∧ q ∈ R ⊕ I = H. (1.5) Surprising as it seems nowadays, it was not until the 1880’s that Josiah Willard Gibbs (1839-1903), a professor at Yale University, argued that the scalar and vector products

7 had their own meaning, independent from their quaternionic origins. It was he who introduced the familiar notation p · q and p ∧ q — before his time these were written ‘−Spq’ and ‘V pq’, indicating the ‘scalar’ and ‘vector’ parts of the quaternionic product pq ∈ H. Another crucial part of vector analysis which originated with Hamilton and the quaternions is the ‘nabla’ operator ∇ (so-called by Hamilton because the symbol ∇ is similar in shape to the Hebrew musical instrument of that name). The familiar gra- 3 dient operator acting on a real differentiable function f(x1, x2, x3): R → R was first written as ∂f ∂f ∂f ∇f := i1 + i2 + i3. (1.6) ∂x1 ∂x2 ∂x3 Hamilton went on to consider applying the operator ∇ to a ‘differentiable quaternion field’ F (x1, x2, x3) = f1(x1, x2, x3)i1 + f2(x1, x2, x3)i2 + f3(x1, x2, x3)i3, obtaining the equation ∂f1 ∂f2 ∂f3 ∂f3 ∂f2 ∂f1 ∂f3 ∂f2 ∂f1 ∇F = − + + + − i1 + − i2 + − i3, ∂x1 ∂x2 ∂x3 ∂x2 ∂x3 ∂x3 ∂x1 ∂x1 ∂x2 which we recognise in modern terminology as

∇F = − div F + curl F.

Applying the ∇ operator to Equation (1.6) leads to the well-known Laplacian operator on 3: R 2 2 2 2 ∂ f ∂ f ∂ f ∇ f = − 2 + 2 + 2 =: ∆f. ∂x1 ∂x2 ∂x3 3 Having obtained the standard scalar product on R , we can obtain the Euclidean 4 metric on R in a similar fashion by relaxing the restriction in Equation (1.5) that p and q should be imaginary. For any p, q ∈ H, we have

Re(pq¯) = p0q0 + p1q1 + p2q2 + p3q3 ∈ R, and so we deﬁne the canonical scalar product on H by hp, qi = Re(pq¯). (1.7)

We deﬁne the norm of a quaternion q ∈ H in the obvious way, putting √ |q| = qq.¯ (1.8)

The norm function is multiplicative, i.e. |pq| = |p||q| for p, q ∈ H. As with complex numbers, we can combine the norm function with conjugation to obtain the inverse of q ∈ H \{0} (it is easily veriﬁed that q−1 =q/ ¯ |q|2 is the unique quaternion such that qq−1 = q−1q = 1). Another quaternionic formula, similar to Equation (1.7), is the identity

Re(pq) = p0q0 − p1q1 − p2q2 − p3q3 ∈ R. (1.9)

8 If we regard p and q as four-vectors in spacetime with the ‘time-axis’ identified with R ⊂ H and the ‘spatial part’ identified with I, this is identical to the Lorentz metric of special relativity. Let a and b be quaternions of unit norm, and consider the involution fa,b : H → H 4 4 given by fa,b(q) = aqb. Then |fa,b(q)| = |q| and the function fa,b : R → R is a rotation. Clearly, f−a,−b = fa,b, so each of these choices for the pair a, b gives the same rotation. 4 In 1855, Cayley showed that all rotations of R can be written in this fashion and that the two possibilities given above are the only two which give the same rotation. Also, ¯ all reflections can be obtained by the involutions fa,b(q) = aqb¯ . 4 One of the beauties of this system is that having obtained all rotations of R , we 3 obtain the rotations of R simply by putting a = b−1, giving the inner automorphism of H, q 7→ aqa−1. (By Cayley’s theorem, all real-algebra automorphisms of H take this form.) This fixes the real line R and rotates the imaginary quaternions I. Identifying I 3 3 with R , the map q 7→ aqa−1 is a rotation of R . According to Cayley, within a year 3 of the discovery of the quaternions Hamilton was aware that all rotations of R can be expressed in this fashion. These discoveries provided much insight into the classical groups SO(3) and SO(4), and helped to develop our knowledge of transformation groups in general. There are interesting questions which arise. Why do the unit quaternions turn out to be so important? In view of the quaternionic version of the Lorentz metric in Equation (1.9), can we use quaternions to write Lorentz tranformations as elegantly? Are there other spaces which might lend themselves to quaternionic treatment? These questions are best addressed using the theory of Lie groups, which the pioneering work of Hamilton and Cayley helped to develop.

1.2 The Lie Group Sp(1) and its Representations

The unit quaternions form a subgroup of H under multiplication, which we call Sp(1). Its importance to the quaternions is equivalent to that of the circle group U(1) of unit complex numbers in complex analysis. As a manifold Sp(1) is the 3-sphere S3. The multiplication and inverse maps are smooth, so Sp(1) is a compact Lie group. The isomorphism ι : H → H ⊂ Mat(2, C) of Equation (1.4) maps Sp(1) isomorphically onto SU(2). The Lie algebra sp(1) of Sp(1) is generated by the elements I, J and K, the Lie bracket being given by the relations [I,J] = 2K,[J, K] = 2I and [K,I] = 2J. We can write these using the matrices of Equation (1.4): i 0 0 1 0 i I = J = K = . 0 −i −1 0 i 0

The complexiﬁcation of sp(1) is the Lie algebra sl(2, C). This is generated over C by the elements 1 0 0 1 0 0 H = X = Y = , (1.10) 0 −1 0 0 1 0 and the relations [H,X] = 2X, [H,Y ] = −2Y and [X,Y ] = H. (1.11)

9 Hence the equations

I = iH, J = X − Y and K = i(X + Y ) (1.12) ∼ give one possible identiﬁcation sp(1) ⊗R C = sl(2, C). For any Q ∈ sp(1), the normaliser N(Q) of Q is deﬁned to be

N(Q) = {P ∈ sp(1) : [P,Q] = 0} = hQi, which is a Cartan subalgebra of sp(1). Identifying a unit vector Q = aI+bJ+cK ∈ sp(1) 2 with the corresponding imaginary quaternion q = ai2 + bi2 + ci3 ∈ S , the exponential map exp : sp(1) → Sp(1) maps hQi to the unit circle in Cq, which we will call U(1)q. In the previous section it was shown that rotations in three and four dimensions can be written in terms of unit quaternions. This is because there is a commutative diagram of Lie group homomorphisms

Sp(1) ∼= Spin(3) ,→ Sp(1) × Sp(1) ∼= Spin(4) ↓ ↓ (1.13) SO(3) ,→ SO(4).

The horizontal arrows are inclusions, and the vertical arrows are 2 : 1 coverings with kernels {1, −1} and {(1, 1), (−1, −1)} respectively. The applications of these homomorphisms in Riemannian geometry are described by Salamon in [S1]. From this, we can see clearly why three- and four-dimensional Euclidean geometry ﬁt so well in a quaternionic framework. We can also see why the same is not true for Lorentzian geometry. Here the important group is the Lorentz group SO(3,1). Whilst there is a double cover SL(2, C) → SO0(3, 1), this does not restrict to a suitably interesting real homomorphism Sp(1)→ SO0(3,1). It is possible to write Lorentz transformations using quaternions (for a modern example see [dL]), but the author has found no way which is simple enough to be really eﬀective. There have been attempts to use the bi- ∼ quaternions {p + iq : p + q ∈ H} = H ⊗R C to apply quaternions to special relativity [Sy], but the mental gymnastics involved in using four ‘square roots of −1’ are cumbersome: the equivalent description of ‘spin transformations’ using the matrices of the group SL(2,C) are a more familiar and fertile ground.

1.2.1 The Representations of Sp(1) A representation of a Lie group G on a vector space V is a Lie group homomorphism ρ : G → GL(V ). We will sometimes refer to V itself as a representation where the map ρ is understood. The diﬀerential dρ is a Lie algebra representation dρ : g → End(V ). The representations of Sp(1) ∼= SU(2) will play an important part in this thesis. Their theory is well-known and used in many situations. Good introductions to Sp(1)- representations include [BD, §2.5] (which describes the action of the group SU(2)) and [FH, Lecture 11] (which provides a description in terms of the action of the Lie algebra sl(2, C)). We recall those points which will be of particular importance. Because it is a compact group, every representation of Sp(1) on a complex vector space V can be written as a direct sum of irreducible representations. The multiplicity

10 of each irreducible in such a decomposition is uniquely determined. Moreover, there is a n unique irreducible representation on C for every n > 0. This makes the representations of Sp(1) particularly easy to describe. Let V1 be the basic representation of SL(2,C) 2 on C given by left-action of matrices upon column vectors. This coincides with the ∼ 2 basic representation of Sp(1) by left-multiplication on H = C . The unique irreducible n+1 th representation on C is then given by the n symmetric power of V1, so we deﬁne

n Vn = S (V1).

The representation Vn is irreducible [BD, Proposition 5.1], and every irreducible representation of Sp(1) is of the form Vn for some nonnegative n ∈ Z [BD, Proposition 5.3]. 2 Let x and y be a basis for C , so that V1 = hx, yi. Then

n n n−1 n−2 2 2 n−2 n−1 n Vn = S (V1) = hx , x y, x y ,..., x y , xy , y i.

The action of SL(2,C) on Vn is given by the induced action on the space of homogeneous polynomials of degree n in the variables x and y. Each of the Lie group representations Vn is a representation of the Lie algebras sl(2, C) and sp(1). Another very important way to describe the structure of these representations is obtained by decomposing them into weight spaces (eigenspaces for the action of a Cartan subalgebra). In terms of H, X and Y , the sl(2, C)-action on V1 is given by H(x) = x X(x) = 0 Y (x) = y (1.14) H(y) = −y X(y) = x Y (y) = 0.

4 To obtain the induced action of sl(2, C) on Vn we use the Leibniz rule A(a · b) = A(a) · b + a · A(b). This gives

H(xn−kyk) = (n − 2k)(xn−kyk) X(xn−kyk) = k(xn−k+1yk−1) (1.15) Y (xn−kyk) = (n − k)(xn−k−1yk+1).

n−k k Each subspace hx y i ⊂ Vn is therefore a weight space of the representation Vn, and the weights are the integers

{n, n − 2, . . . , n − 2k, . . . , 2 − n, −n}.

Thus Vn is also characterised by being the unique irreducible representation of sl(2, C) with highest weight n. We can compute the action of sp(1) on Vn by substituting I, J and K for H, X and Y using the relations of (1.12). Another important operator which acts on an sp(1)-representation is the Casimir operator C = I2 + J 2 + K2 = −(H2 + 2XY + 2YX). It is easy to show that for any n−k k x y ∈ Vn, C(xn−kyk) = −n(n + 2)xn−kyk. (1.16)

4This can be found in [FH, p. 110], which describes the action of Lie groups and Lie algebras on tensor products.

11 Each irreducible representation Vn is thus an eigenspace of the Casimir operator with eigenvalue −n(n + 2). Let Vm and Vn be two Sp(1)-representations. Then their tensor product Vm ⊗ Vn is naturally an Sp(1) × Sp(1)-representation, 5 and also a representation of the diagonal Sp(1)-subgroup, the action of which is given by

g(u ⊗ v) = g(u) ⊗ g(v).

The irreducible decomposition of the diagonal Sp(1)-representation on Vm ⊗ Vn is given by the famous Clebsch-Gordon formula, ∼ Vm ⊗ Vn = Vm+n ⊕ Vm+n−2 ⊕ · · · ⊕ Vm−n+2 ⊕ Vm−n for m ≥ n. (1.17)

This can be proved using characters [BD, Proposition 5.5] or weights [FH, Exercise 11.11].

Real and quaternionic representations It is standard practice to work primarily with representations on complex vector spaces. Representations on real (and quaternionic) vector spaces are obtained using antilinear structure maps. A thorough guide to this process is in [BD, §2.6]. In the case of Sp(1)- representations, we deﬁne the structure map σ1 : V1 → V1 by

σ1(z1x + z2y) = −z¯2x +z ¯1y z1, z2 ∈ C.

2 Then σ1 = −1 , and σ1 coincides with the map j of Section 1.1.2. Let σn be the map which σ1 induces on Vn, i.e.

n−k k k k n−k σn(z1x y ) = (−1) z¯1x y . (1.18)

2 σ If n = 2m is even then σ2m = 1 and σ2m is a real structure on V2m. Let V2m be the set σ ∼ 2m+1 of ﬁxed-points of σ2m. Then V2m = R is preserved by the action of Sp(1), and σ ∼ V2m ⊗R C = V2m.

σ 2m+1 Thus V2m is a representation of Sp(1) on the real vector space R . 2 If on the other hand n = 2m − 1 is odd, σ2m−1 = −1. Then Sp(1) acts on the un- 4m derlying real vector space R . This real vector space comes equipped with the complex structure i and the structure map σ2m−1, in such a way that the subspace

∼ 4 hv, iv, σ2m−1(v), iσ2m−1(v)i = R ∼ m is isomorphic to the quaternions; thus V2m−1 = H . This is why a complex antilinear map σ on a complex vector space V such that σ2 = −1 is called a quaternionic structure.

5Since Sp(1) × Sp(1) =∼ Spin(4), we can construct all Spin(4) and hence all SO(4) representations in this fashion — see [S1, §3].

12 1.3 Diﬃculties with the Quaternions

Quaternions are far less predictable than their lower-dimensional cousins, due to the great complicating factor of non-commutativity. Many of the ideas which work beautifully for real or complex numbers are not suited to quaternions, and attempts to use them often result in lengthy and cumbersome mathematics — most of which dates from the 19th century and is now almost forgotten. However, with modesty and care, quaternions can be used to recreate many of the structures over the real and complex numbers with which we are familiar. The purpose of this section is to outline some of the diﬃculties with a few examples, which highlight the need for caution: but also, it is hoped, point the way to some of the successes we will encounter.

1.3.1 The Fundamental Theorem of Algebra for Quaternions The single biggest reason for using the complex numbers in preference to any other number ﬁeld is the so-called ‘Fundamental Theorem of Algebra’ — every complex polynomial of degree n has precisely n zeros, counted with multiplicities. The real numbers are not so well-behaved: a real polynomial of degree n can have fewer than n real roots. Quaternion behaviour is many degrees freer and less predictable. If we multiply together two ‘linear factors’, we obtain the following expression:

(a1X + b1)(a2X + b2) = a1Xa2X + a1Xb2 + b1a2X + b1b2.

It quickly becomes obvious that non-commutativity is going make any attempt to factorise a general polynomial extremely troublesome. Moreover, there are many polynomials which display extreme behaviour. For ex- 2 2 2 2 ample, the cubic X i1Xi1 + i1X i1X − i1Xi1X − Xi1X i1 takes the value zero for all X ∈ H. At the other extreme, since i1X − Xi1 ∈ I for all X ∈ H, the equation i1X − Xi1 + 1 = 0 has no solutions at all! In order to arrive at any kind of ‘fundamental theorem of algebra’, we need to restrict our attention considerably. We deﬁne a monomial of degree n to be an expression of the form a0Xa1Xa2 ··· an−1Xan, ai ∈ H \{0}. Then there is the following ‘fundamental theorem of algebra for quaternions’:

Theorem 1.3.1 [EKR, p. 205] Let f be a polynomial over H of degree n > 0 of the form m + g, where m is a monomial of degree n and g is a polynomial of degree < n. Then the mapping f : H → H is surjective, and in particular f has zeros in H.

This is typical of the diﬃculties we encounter with quaternions. There is a quaternionic analogue of the theorems for real and complex numbers, but because of non- commutativity the quaternionic version is more complicated, less general and because of this less useful.

13 1.3.2 Calculus with Quaternions The monumental successes of complex analysis make it natural to look for a similar theory for quaternions. Complex analysis can be described as the study of holomorphic functions. A function f : C → C is holomorphic if it has a well-defined complex derivative. One of the fundamental results in complex analysis is that every holomorphic function is analytic i.e. can be written as a convergent power series. Sadly, neither of these definitions proves interesting when applied to quaternions — the former is too restrictive and the latter too general. For a function f : H → H to have a well-defined derivative df = lim(f(q + h) − f(q))h−1, dq h→0 it can be shown [Su, §3, Theorem 1] that f must take the form f(q) = a + bq for some a, b ∈ H, so the only functions which are quaternion-differentiable in this sense are affine linear. In contrast to the complex case, the components of a quaternion can be written as quaternionic polynomials, i.e. for q = q0 + q1i1 + q2i2 + q3i3, 1 1 q0 = (q − i1qi1 − i2qi2 − i3qi3) q1 = (q − i1qi1 + i2qi2 + i3qi3) etc. (1.19) 4 4i1 This takes us to the other extreme: the study of quaternionic power series is the same 4 as the theory of real-analytic functions on R . Hamilton and his followers were aware of this — it was in Hamilton’s work on quaternions that some of the modern ideas in the theory of functions of several real variables first appeared. Despite the benefits of this work to mathematics as a whole, Hamilton never developed a successful ‘quaternion calculus’. It was not until the work of R. Fueter, in 1935, that a suitable definition of a ‘regular quaternionic function’ was found, using a quaternionic analogue of the Cauchy-Riemann equations. A regular function on H is defined to be a real-differentiable function f : H → H which satisfies the Cauchy-Riemann-Fueter equations: ∂f ∂f ∂f ∂f + i1 + i2 + i3 = 0. (1.20) ∂q0 ∂q1 ∂q2 ∂q3 The theory of quaternionic analysis which results is modestly successful, though it is little-known. The work of Fueter is described and extended in the papers of Deavours [D] and Sudbery [Su], which include quaternionic versions of Morera’s theorem, Cauchy’s theorem and the Cauchy integral formula. As usual, non-commutativity causes immediate algebraic difficulties. If f and g are regular functions, it is easy to see that their sum f + g must also be regular, as must the left scalar multiple qf for q ∈ H; but their product fg, the composition f ◦ g and the right scalar multiple fq need not be. Hence regular functions form a left H-module, but it is difficult to see how to describe any further algebraic structure.

1.3.3 Quaternion Linear Algebra In the same way as we deﬁne real or complex vector spaces, we can deﬁne quaternionic vector spaces or H-modules — a real vector space with an H-action, which we shall call

14 scalar multiplication. There is the added complication that we need to say whether this multiplication is on the left or the right. We will work with left H-modules — this choice is arbitrary and has only notational effects on the resulting theory. A left H-module is thus a real vector space U with an action of H on the left, which we write (q, u) 7→ q · u n or just qu, such that p(qu) = (pq)u for all p, q ∈ H and u ∈ U. For example, H is an H-module with the obvious left-multiplication. Several of the familiar ideas which work for vector spaces over a commutative field work just as well for H-modules. For example, we can define quaternion linear maps between H-modules in the obvious way, and so we can define a dual H-module U × of quaternion linear maps α : U → H. However, if we try to define quaternion bilinear maps we run into trouble. If A, B and C are vector spaces over the commutative field F, then an F-bilinear map µ : A×B → C satisfies µ(f1a, b) = f1µ(a, b) and µ(a, f2b) = f2µ(a, b) for all a ∈ A, b ∈ B, f1, f2 ∈ F. This is equivalent to having µ(f1a, f2b) = f1f2µ(a, b). Now suppose that U, V and W are (left) H-modules. Let q1, q2 ∈ H and let u ∈ U, v ∈ V . If we try to define a bilinear map µ : U × V → W as above, then we need both µ(q1u, q2v) = q1µ(u, q2v) = q2q1µ(u, v) and µ(q1u, q2v) = q2µ(q1u, v) = q1q2µ(u, v). Since q1q2 6= q2q1 in general, this cannot work. Similar difficulties arise if we try to define a tensor product over the quaternions. If A and B are vector spaces over the commutative field F, their tensor product over F is defined by

F (A, B) A ⊗ B = , (1.21) F R(A, B) where F (A, B) is the vector space freely generated (over F) by all elements (a, b) ∈ A×B and R(A, B) is the subspace of F (A, B) generated by all elements of the form

(a1 + a2, b) − (a1, b) − (a2, b)(a, b1 + b2) − (a, b1) − (a, b2)

f(a, b) − (f(a), b) and f(a, b) − (a, f(b)), for a, aj ∈ A, b, bj ∈ B and f ∈ F. Not surprisingly, this process does not work for quaternions. Let U and V be left H-modules. Then if we deﬁne the ideal R(U, V ) in the same way as above, we discover that R(U, V ) is equal to the whole of F (U, V ), so

U ⊗F V = {0}. One way around both of these difficulties is to demand that our H-modules should also have an H-action on the right. We can now define an H-bilinear map to be one which satisfies µ(q1u, vq2) = q1µ(u, v)q2, or alternatively µ(q1u, q2v) = q1µ(u, v)¯q2. Similarly, for our tensor product we can define a generator uq ⊗ v − u ⊗ qv (replacing f(u) ⊗ v − u ⊗ f(v)) for R. In this (more restricted) case we do obtain a well-defined ‘quaternionic tensor product’ U⊗HV = F (U, V )/R(U, V ) which inherits a left H-action from U and a right H-action from V . The drawback with this system is that it does not really provide any new insights. If we insist on having a left and a right H-action, we restrict ourselves to talking about n n H-modules of the form H = H ⊗R R . In this case, we do get the useful relation m n ∼ mn m n ∼ mn H ⊗HH = H , but this is only saying that H⊗RR ⊗RR = H⊗RR . In this context, our ‘quaternionic tensor product’ is merely a real tensor product in a quaternionic setting.

15 1.3.4 Summary By now we have become familiar with some of the more elementary ups and downs of the quaternions. In the 20th century they have often been viewed as a sort of mathematical Cinderella, more recent techniques being used to describe phenomena which were ﬁrst thought to be profoundly quaternionic. However, we have seen that it is possible to produce quaternionic analogues of some of the most basic algebraic and analytic ideas of real and complex numbers, often with interesting and useful results. In the next chapter we will continue to explore this process, turning our attention to quaternionic structures in diﬀerential geometry.

16 Chapter 2

Quaternionic Diﬀerential Geometry

In this chapter we review some of the ways in which quaternions are used to define geometric stuctures on differentiable manifolds. In the same way that real and complex n n manifolds are modelled locally by the vector spaces R and C respectively, there are n manifolds which can be modelled locally by the H-module H . These models work by defining tensors whose action on the tangent spaces to a manifold is the same as the action of the quaternions on an H-module. There are two important classes of manifolds which we shall consider: those which are called ‘quaternionic manifolds’, and a more restricted class called ‘hypercomplex manifolds’. In this chapter we describe these important geometric structures. We also review the decomposition of exterior forms on complex manifolds, and examine some of the parallels of this theory which have already been found in quaternionic geometry. Many of the algebraic and geometric foundations of the material in this chapter are collected in (or can be inferred from) Fujiki’s comprehensive article [F].

2.1 Complex, Hypercomplex and Quaternionic Man- ifolds

Complex Manifolds A complex manifold is a 2n-dimensional real manifold M which admits an atlas of n complex charts, all of whose transition functions are holomorphic maps from C to itself. As we saw in Section 1.3, the simplest notions of a ‘quaternion-differentiable map from H to itself’ are either very restrictive or too general, and the ‘regular functions’ of Fueter and Sudbery are not necessarily closed under composition. This makes the notion of ‘quaternion-differentiable transition functions’ less interesting that one might hope. An equivalent way to define a complex manifold is by the existence of a special tensor called a complex structure. A complex structure on a real vector space V is an 2 automorphism I : V → V such that I = − idV . (It follows that dim V is even.) The ∼ n complex structure I gives an isomorphism V = C , since the operation ‘multiplication n by i’ defines a standard complex structure on C . An almost complex structure on a 2n-dimensional real manifold M is a smooth tensor I ∈ C∞(End(TM)) such that I is a complex structure on each of the fibres TxM.

17 n Now, if M is a complex manifold, each tangent space TxM is isomorphic to C , so taking I to be the standard complex structure on each TxM deﬁnes an almost complex structure on M. An almost complex stucture I which arises in this way is called a complex structure on M, in which case I is said to be integrable. The famous Newlander- Nirenberg theorem states that an almost complex structure I is integrable if and only if the Nijenhuis tensor of I

NI (X,Y ) = [X,Y ] + I[IX,Y ] + I[X,IY ] − [IX,IY ]

∞ vanishes for all X,Y ∈ C (TM), for all x ∈ M. The Nijenhuis tensor NI measures the (0, 1)-component of the Lie bracket of two vector fields of type (1, 0). 1 On a complex manifold the Lie bracket of two (1, 0)-vector fields must also be of type (1, 0). Thus if I is an almost complex structure on M and NI ≡ 0, then M is a complex manifold in the sense of the first definition given above. We can talk about the complex manifold (M,I) if we wish to make the extra geometric structure more explicit — especially as the manifold M might admit more than one complex structure.

Hypercomplex Manifolds This way of defining a complex manifold adapts itself well to the quaternions. A hypercomplex structure on a real vector space V is a triple (I, J, K) of complex structures on V satisfying the equation IJ = K. (It follows that dim V is divisible by 4.) If we identify I, J and K with left-multiplication by i1, i2 and i3, a hypercomplex structure gives ∼ n an isomorphism V = H . Equivalently, a hypercomplex structure is defined by a pair of complex structures I and J with IJ = −JI. It is easy to see that if (I, J, K) is a hypercomplex structure on V , then each element of the set {aI+bJ+cK : a2+b2+c2 = 1} ∼= S2 is also a complex structure. We arrive at the following quaternionic version of a complex manifold: Definition 2.1.1 An almost hypercomplex structure on a 4n-dimensional manifold M is a triple (I, J, K) of almost complex structures on M which satisfy the relation IJ = K. If all of the complex structures are integrable then (I, J, K) is called a hypercomplex structure on M, and M is a hypercomplex manifold.

∼ n A hypercomplex structure on M defines an isomorphism TxM = H at each point x ∈ M. As on a vector space, a hypercomplex structure on a manifold M defines a 2-sphere S2 of complex structures on M. Some choices are inherent in this standard definition. A hypercomplex structure as defined above gives TM the structure of a left H-module, since IJ = K. This induces a right H-module structure on T ∗M, using the standard definition hI(ξ),Xi = hξ, I(X)i etc., for all ξ ∈ T ∗M,X ∈ TM, since on T ∗M we now have

hξ, IJ(X)i = hI(ξ),J(X)i = hJI(ξ),Xi.

In this thesis we will make more use of the hypercomplex structure on T ∗M than that on TM. Because of this we will usually deﬁne our hypercomplex structures so that IJ = K on T ∗M rather than TM. This has only notational eﬀect on the theory, but it does pay to

1Tensors of type (p, q) are deﬁned in the next section.

18 be aware of how it aﬀects other standard notations. For example, for us the hypercomplex − structure acts trivially on the anti-self-dual 2-forms ω1 = dx0 ∧ dx1 − dx2 ∧ dx3 etc. This encourages us to think of a connection whose curvature is acted upon trivially by the hypercomplex structure as anti-self-dual rather than self-dual, whereas some authors use the opposite convention. Such things are largely a matter of taste — we are choosing to follow the notation of Joyce [J1] for whom regular functions form a left H-module, whereas Sudbery’s regular functions form a right H-module. One important diﬀerence between complex and hypercomplex geometry is the existence of a special connection. A complex manifold generally admits many torsion-free connections which preserve the complex structure. By contrast, on a hypercomplex manifold there is a unique torsion-free connection ∇ such that

∇I = ∇J = ∇K = 0.

This was proved by Obata in 1956, and the connection ∇ is called the Obata connection. Complex and hypercomplex manifolds can be decribed succinctly in terms of G- structures on manifolds. Let P be the principal frame bundle of M, i.e. the GL(n, R)- ∼ 4n bundle whose ﬁbre over x ∈ M is the group of isomorphisms TxM = R . Let G be a Lie subgroup of GL(n, R). A G-structure Q on M is a principal subbundle of P with structure group G. 2n Suppose M has an almost complex structure. The group of automorphisms of TxM preserving such a structure is isomorphic to GL(n, C). Thus an almost complex structure I and a GL(n, C)-structure Q on M contain the same information. The bundle Q admits a torsion-free connection if and only if there is a torsion-free linear connection ∇ on M with ∇I = 0, in which case it is easy to show that I is integrable. Thus a complex manifold is precisely a real manifold M 2n with a GL(n, C)-structure Q admitting a torsion-free connection (in which case Q itself is said to be ‘integrable’). If M 4n has an almost hypercomplex structure then the group of automorphisms preserving this structure is isomorphic to GL(n, H). Following the same line of argument, a hypercomplex manifold is seen to be a real manifold M with an integrable GL(n, H)- structure Q. The uniqueness of any torsion-free connection on Q follows from analysing the Lie algebra gl(n, H). This process is described in [S3, §6].

Quaternionic Manifolds and the Structure Group GL(1, H)GL(n, H) Not all of the manifolds which we wish to describe as ‘quaternionic’ admit hypercomplex 1 structures. For example, the quaternionic projective line HP is diﬀeomorphic to the 4-sphere S4. It is well-known that S4 does not even admit a global almost complex 1 structure; so HP can certainly not be hypercomplex, despite behaving extremely like the quaternions locally. The reason (and the solution) for this diﬃculty is that GL(n, H) is not the largest subgroup of GL(4n, R) preserving a quaternionic structure. If we think of GL(n, H) n as acting on H by right-multiplication by n × n quaternionic matrices, then the action of GL(n, H) commutes with that of the left H-action of the group GL(1, H). Thus n ∗ the group of symmetries of H is the product GL(1, H) ×R GL(n, H) , which we write GL(1, H)GL(n, H). We can multiply on the right by any real multiple of the identity

19 ∗ (since GL(1, H) and GL(n, H) share the same centre R ), so without loss of generality we can reduce the first factor to Sp(1). Thus GL(1, H)GL(n, H) is the same as Sp(1)GL(n, H) = Sp(1) ×Z2 GL(n, H). Definition 2.1.2 [S3, 1.1] A quaternionic manifold is a 4n-dimensional real manifold M (n ≥ 2) with an Sp(1)GL(n, H)-structure Q admitting a torsion-free connection. When n = 1 the situation is different, since Sp(1)×Sp(1) ∼= SO(4). In four dimensions we make the special definition that a quaternionic manifold is a self-dual conformal manifold. In terms of tensors, quaternionic manifolds are a generalisation of hypercomplex manifolds in the following way. Each tangent space TxM still admits a hypercomplex ∼ n structure giving an isomorphism TxM = H , but this isomorphism does not necessarily arise from globally defined complex structures on M. There is still an S2-bundle of local almost-complex structures which satisfy IJ = K, but it is free to ‘rotate’. For a comprehensive study of quaternionic manifolds see [S3] and Chapter 9 of [S4].

Riemannian Manifolds in Complex and Quaternionic Geometry

Suppose the GL(n, C)-structure Q on a complex manifold M admits a further reduc- tion to an integrable U(n)-structure Q0. Then M admits a Riemannian metric g with g(IX,IY ) = g(X,Y ) for all X,Y ∈ TxM for all x ∈ M. We also define the differentiable 2-form ω(X,Y ) = g(IX,Y ). If such a metric arises from an integrable U(n)-structure Q0 then ω will be a closed 2-form, and M is a symplectic manifold — so M has compatible complex and symplectic structures. In this case M is called a Kählermanifold; an integrable U(n)-structure is called a Kählerstructure; the metric g is called a Kähler metric and the symplectic form ω is called a Kählerform. The quaternionic analogue of the compact group U(n) is the group Sp(n). A hypercomplex manifold whose GL(n, H)-structure Q reduces to an integrable Sp(n)-structure Q0 admits a metric g which is simultaneously Kählerfor each of the complex structures I,J and K. Such a manifold is called hyperkähler . Using each of the complex structures, we define three independent symplectic forms ωI , ωJ and ωK . Then the complex 2-form ωJ + iωK is holomorphic with respect to the complex structure I, and a hyperkähler manifold has compatible hypercomplex and complex-symplectic structures. Hyperkähler manifolds are studied in [HKLR], which gives a quotient construction for hyperkähler manifolds. Similarly, if a quaternionic manifold has a metric compatible with the torsion-free Sp(1)GL(n, H)-structure, then the Sp(1)GL(n, H)-structure Q reduces to an Sp(1)Sp(n)- structure Q0 and M is said to be quaternionic Kähler. The group Sp(1)Sp(n) is a maximal proper subgroup of SO(4n) except when n = 1, where as we know Sp(1)Sp(1) = SO(4). In four dimensions a manifold is said to be quaternionic Kähler if and only if it is self-dual and Einstein. Quaternionic Kählermanifolds are the subject of [S2].

2.2 Diﬀerential Forms on Complex Manifolds

This section consists of background material in complex geometry, especially ideas which encourage the development of interesting quaternionic versions. More information on this

20 and other aspects of complex geometry can be found in [W] and [GH, §0.2]. Let (M,I) be a complex manifold. Then I gives TM and T ∗M the structure of a

U(1)-representation and the complexiﬁcation C⊗R TM splits into two weight spaces with ∗ weights ±i. The same is true of C ⊗R T M. There are various ways of writing these ∗ ∗ ∗ weight spaces; fairly standard is the notation C ⊗R T M = T1,0M ⊕ T0,1M, where these summands are the +i and −i eigenspaces of I respectively. However, for our purposes it will be more useful to adopt the notation of [S3], writing

∗ 1,0 0,1 C ⊗R T M = Λ M ⊕ Λ M, (2.1)

1,0 ∗ 0,1 ∗ ∞ so Λ M = T1,0M and Λ M = T0,1M. A holomorphic function f ∈ C (M, C) is a smooth function whose derivative takes values only in Λ1,0M for all m ∈ M, i.e. f is holomorphic if and only if df ∈ C∞(M, Λ1,0M). A closely linked statement is that Λ1,0M is a holomorphic vector bundle. Thus Λ1,0M is called the holomorphic cotangent space and Λ0,1M is called the antiholomorphic cotangent space of M. If we reverse the sense of the complex structure (i.e. if we swap I for −I) then we reverse the roles of the holomorphic and antiholomorphic spaces, which is why a function which is holomorphic with respect to I is antiholomorphic with respect to −I. From standard multilinear algebra, the decomposition (2.1) gives rise to a decomposition of exterior forms of all powers

k k ∗ M p ∗ k−p ∗ C ⊗R Λ T M = Λ (T1,0M) ⊗ Λ (T0,1M). p=0

Deﬁne the bundle p,q p ∗ q ∗ Λ M = Λ T1,0M ⊗ Λ T0,1M. (2.2) With this notation, Equation (2.1) is an example of the more general decomposition into types k ∗ M p,q C ⊗R Λ T M = Λ M. (2.3) p+q=k A smooth section of the bundle Λp,qM is called a diﬀerential form of type (p, q) or just a (p, q)-form. We write Ωp,q(M) for the set of (p, q)-forms on M, so M Ωp,q(M) = C∞(M, Λp,qM) and Ωk(M) = Ωp,q(M). p+q=k

Define two first-order differential operators,

∂ :Ωp,q(M) → Ωp+1,q(M) ∂ :Ωp,q(M) → Ωp,q+1(M) and (2.4) ∂ = πp+1,q ◦ d ∂ = πp,q+1 ◦ d, where πp,q denotes the natural projection from C ⊗ ΛkM onto Λp,qM. The operator ∂ is called the Dolbeault operator. These deﬁnitions rely only on the fact that I is an almost complex structure. A crucial fact is that if I is integrable, these are the only two components represented in

21 Figure 2.1: The Dolbeault Complex

Ω2,0 ...... etc. ¨¨* HH ∂¨ H ¨¨ HH ¨ ∂ Hj Ω1,0 ¨* H ¨* ∂ ¨ H ∂ ¨ ¨¨ HH ¨¨ ¨¨ ∂ HHj ¨¨ Ω0,0 = C∞(M, C) Ω1,1 H ¨* H H ∂ ¨ H HH ¨¨ HH ∂ HHj ¨¨ ∂ HHj Ω0,1 H ¨* HH ∂¨¨ H ¨ ∂ HHj ¨¨ Ω0,2 ...... etc. the exterior diﬀerential d, i.e. d = ∂ + ∂ [WW, Proposition 2.2.2, p.105]. An immediate consequence of this is that on a complex manifold M,

2 ∂2 = ∂∂ + ∂∂ = ∂ = 0. (2.5)

p,q This gives rise to the Dolbeault complex. Writing ∂ for the particular map ∂ : Ωp,q → Ωp,q+1, we deﬁne the Dolbeault cohomology groups

p,q p,q Ker(∂ ) H = p,q−1 . ∂ Im(∂ )

A function f ∈ C∞(M, C) is holomorphic if and only if ∂f = 0 and for this reason ∂ is sometimes called the Cauchy-Riemann operator. Similarly, a (p, 0)-form α is said to be holomorphic if and only if ∂α = 0. A useful way to think of the Dolbeault complex is as a decomposition of C ⊗ ΛkT ∗M into types of U(1)-representation. The representations of U(1) on complex vector spaces are particularly easy to understand. Since U(1) is abelian, the irreducible representations all are one-dimensional. They are parametrised by the integers, taking the form

∗ iθ niθ %n : U(1) → GL(1, C) = C %n : e → e for some n ∈ Z. Representations of the Lie algebra u(1) are then of the form d%n : z → nz. The Dolbeault complex begins with the representation of the complex structure ∼ ∗ • ∗ hIi = u(1) on C ⊗ T M. This induces a representation on C ⊗ Λ T M. It is easy to see that if ω ∈ Λp,qM, I(ω) = i(p − q)ω. In other words, Λp,qM is the bundle of U(1)-representations of the type %p−q.

22 2.3 Diﬀerential Forms on Quaternionic Manifolds

Any G-structure on a manifold M induces a representation of G on the exterior algebra of M. Fujiki’s account of this [F, §2] explains many quaternionic analogues of complex and Kählergeometry. The decomposition of differential forms on quaternionic Kählermanifolds began by considering the fundamental 4-form

Ω = ωI ∧ ωI + ωJ ∧ ωJ + ωK ∧ ωK , where ωI , ωJ and ωK are the local Kählerforms associated to local almost complex structures I,J and K with IJ = K. The fundamental 4-form is globally defined and invariant under the induced action of Sp(1)Sp(n) on Λ4T ∗M. Kraines [Kr] and Bonan [Bon] used the fundamental 4-form to decompose the space ΛkT ∗M in a similar way to the Lefschetz decomposition of differential forms on a Kähler manifold [GH, p. 122]. A differential k-form µ is said to be effective if Ω∧∗µ = 0, where ∗ :ΛkT ∗M → Λ4n−kT ∗M is the Hodge star. This leads to the following theorem:

Theorem 2.3.1 [Kr, Theorem 3.5][Bon, Theorem 2] For k ≤ 2n + 2, every every k-form φ admits a unique decomposition

X j φ = Ω ∧ µk−4j, 0≤j≤k/4

where the µk−4j are eﬀective (k − 4j)-forms.

Bonan further refines this decomposition for quaternion-valued forms, using exterior multiplication by the globally defined quaternionic 2-form Ψ = i1ωI + i2ωJ + i3ωK . Note that Ψ ∧ Ψ = −2Ω. Another way to consider the decomposition of forms on a quaternionic manifold is as representations of the group Sp(1)GL(n, H). We express the Sp(1)GL(n, H)-representation n on H by writing n ∼ H = V1 ⊗ E, (2.6) 2 where V1 is the basic representation of Sp(1) on C and E is the basic representation 2n of GL(n, H) on C . (This uses the standard convention of working with complex representations, which in the presence of suitable structure maps can be thought of as complexified real representations. In this case, the structure map is the tensor product of the quaternionic structures on V1 and E.) This representation also describes the (co)tangent bundle of a quaternionic manifold in the following way. Let P be a principal G-bundle over the differentiable manifold M and let W be a G-module. We define the associated bundle P × W W = P × W = , G G where g ∈ G acts on (p, w) ∈ P × W by (p, w) · g = (f · g, g−1 · w). Then W is a vector bundle over M with fibre W . We will usually just write W for W, relying on context

23 to distinguish between the bundle and the representation. Following Salamon [S3, §1], if M 4n is a quaternionic manifold with Sp(1)GL(n, H)-structure Q, then the cotangent bundle is a vector bundle associated to the principal bundle Q and the representation V1 ⊗ E, so that we write ∗ C ∼ (T M) = V1 ⊗ E (2.7) (though we will usually omit the complexiﬁcation sign). This induces an Sp(1)GL(n, H)- action on the bundle of exterior k-forms ΛkT ∗M,

[k/2] [k/2] k ∗ ∼ k ∼ M k−2j k ∼ M k Λ T M = Λ (V1 ⊗ E) = S (V1) ⊗ Lj = Vk−2j ⊗ Lj , (2.8) j=0 j=0

k where Lj is an irreducible representation of GL(n, H). This decomposition is given by Salamon [S3, §4], along with more details concerning the nature of the GL(n, H) k k ∗ representations Lj . If M is quaternion Kähler, Λ T M can be further decomposed into representations of the compact group Sp(1)Sp(n). This refinement is performed in detail by Swann [Sw], and used to demonstrate significant results. If we symmetrise completely on V1 in Equation (2.8) to obtain Vk, we must antisym- metrise completely on E. Salamon therefore defines the irreducible subspace

k ∼ k A = Vk ⊗ Λ E. (2.9)

The bundle Ak can be described using the decomposition into types for the local almost complex structures on M as follows [S3, Proposition 4.2]: 2

k X k,0 A = ΛI M. (2.10) I∈S2

Letting p denote the natural projection p :ΛkT ∗M → Ak and setting D = p ◦ d, we deﬁne a sequence of diﬀerential operators

0 −→ C∞(A0) −→D=d C∞(A1 = T ∗M) −→D C∞(A2) −→D ... −→D C∞(A2n) −→ 0. (2.11)

This is accomplished using only the fact that M has an Sp(1)GL(n, H)-structure; such a manifold is called ‘almost quaternionic’. The following theorem of Salamon relates the integrability of such a structure with the sequence of operators in (2.11):

Theorem 2.3.2 [S3, Theorem 4.1] An almost quaternionic manifold is quaternionic if and only if (2.11) is a complex.

This theorem is analogous to the familiar result in complex geometry that an almost 2 complex structure on a manifold is integrable if and only if ∂ = 0.

2 This is because every Sp(1)-representation Vn is generated by its highest weight spaces taken with respect to all the diﬀerent linear combinations of I, J and K. We will use such descriptions in detail in later chapters, and ﬁnd that they play a prominent role in quaternionic algebra.

24 Chapter 3

A Double Complex on Quaternionic Manifolds

Until now we have been discussing known material. In this chapter we present a discovery which as far as the author can tell is new — a double complex of exterior forms on quaternionic manifolds. We argue that this is the best quaternionic analogue of the Dolbeault complex. The ‘top row’ of this double complex is exactly the complex (2.11) discovered by Simon Salamon, which plays a similar role to that of the (k, 0)-forms on a complex manifold. The new double complex is obtained by simplifying the Bonan decomposition of Equation (2.8). Instead of using the more complicated structure of ΛkT ∗M as an Sp(1)GL(n, H)-module, we consider only the action of the Sp(1)-factor and decompose ΛkT ∗M into irreducible Sp(1)-representations — a fairly easy process achieved by considering weights. The resulting decomposition gives rise to a double complex through ∼ the Clebsch-Gordon formula, in particular the isomorphism Vr ⊗ V1 = Vr+1 ⊕ Vr−1. This encourages us to define two ‘quaternionic Dolbeault’ operators D and D, and leads to new cohomology groups on quaternionic manifolds. The main geometric difference between this double complex and the Dolbeault complex is that whilst the Dolbeault complex is a diamond, our double complex forms an isosceles triangle, as if the diamond is ‘folded in half’. This is more like the decomposition of real-valued forms on complex manifolds. Determining where our double complex is elliptic has been far more difficult than for the de Rham or Dolbeault complexes. The ellipticity properties of our double complex are more similar to those of a real-valued version of the Dolbeault complex. Because of this similarity, we shall begin with a discussion of real-valued forms on complex manifolds.

3.1 Real forms on Complex Manifolds

Let M be a complex manifold and let ω ∈ Λp,q = Λp,qM. Then ω ∈ Λq,p, and so ω + ω p,q q,p is a real-valued exterior form in (Λ ⊕ Λ )R, where the subscript R denotes the fact p,q q,p that we are talking about real forms. The space (Λ ⊕ Λ )R is a real vector bundle associated to the principal GL(n, C)-bundle deﬁned by the complex structure. This gives

25 a decomposition of real-valued exterior forms, 1

M k , k Λk T ∗M = (Λp,q ⊕ Λq,p) ⊕ Λ 2 2 . (3.1) R R R p+q=k p>q

k , k The condition p > q ensures that we have no repetition. The bundle Λ 2 2 only appears R when k is even. It is its own conjugate and so naturally a real vector bundle associated to the trivial representation of U(1). p,q q,p p+1,q q,p+1 Let ω + ω ∈ (Ω ⊕ Ω )R. Then ∂ω + ∂ω ∈ (Ω ⊕ Ω )R. Call this operator ∂ ⊕ ∂. Then d(ω + ω) = (∂ ⊕ ∂)(ω + ω) + (∂ ⊕ ∂)(ω + ω)

p+1,q q,p+1 p,q+1 q+1,p ∈ (Ω ⊕ Ω )R ⊕ (Ω ⊕ Ω )R. When p = q, ω = ω, so ∂ ⊕ ∂ = ∂ ⊕ ∂ = d, and there is only one diﬀerential operator acting on Ωp,p. This gives the following double complex (where the real subscripts are omitted for convenience).

Figure 3.1: The Real Dolbeault Complex

Ω3,0 ⊕ Ω0,3 ...... etc. ¨* H ∂ ⊕ ∂ ¨¨ HH ¨ H ¨¨ ∂ ⊕ ∂ HHj Ω2,0 ⊕ Ω0,2 ¨¨* HH ¨¨* ∂ ⊕ ∂ ¨ H ∂ ⊕ ∂ ¨ ¨¨ HH ¨¨ ¨ ∂ ⊕ ∂ Hj ¨ Ω1,0 ⊕ Ω0,1 Ω2,1 ⊕ Ω1,2 ...... etc. ¨¨* HH ¨¨* HH d ¨ H d ¨ H ¨¨ HH ¨¨ HH ¨ ∂ ⊕ ∂ Hj ¨ ∂ ⊕ ∂ Hj Ω0,0 = C∞(M) Ω1,1

Thus there is a double complex of real forms on a complex manifold, obtained by k ∗ decomposing ΛxT M into subrepresentations of the action of u(1) = hIi, induced from ∗ the action on Tx M. This double complex is less well-behaved than its complex-valued counterpart; in particular, it is not elliptic everywhere. We shall now show what this means, and why it is signiﬁcant.

Ellipticity Whether or not a diﬀerential complex on a manifold is ‘elliptic’ is an important question, with striking topological, algebraic and physical consequences. Examples of elliptic

1 p,q q,p This decomposition is also given by Reyes-Cari´on[R, §3.1], who calls the bundle (Λ ⊕ Λ )R [[Λp,q]].

26 complexes include the de Rham and Dolbeault complexes. A thorough description of elliptic operators and elliptic complexes can be found in [W, Chapter 5]. Let E and F be vector bundles over M, and let Φ : C∞(E) → C∞(F ) be a differential operator. (We will be working with first-order operators, and so will only describe this ∗ case.) At every point x ∈ M and for every nonzero ξ ∈ Tx M, we define a linear map ∞ σΦ(x, ξ): Ex → Fx called the principal symbol of Φ, as follows. Let ∈ C (E) with (x) = e, and let f ∈ C∞(M) with f(x) = 0, df(x) = ξ. Then

σΦ(x, ξ)e = Φ(f)|x.

∂ In coordinates, σ is often found by replacing the operator ∂xj with exterior multiplication j ∂ by a cotangent vector ξ dual to ∂xj . For example, for the exterior differential d : k k+1 Ω (M) → Ω (M) we have σd(x, ξ)ω = ω ∧ ξ. The operator Φ is said to be elliptic at ∗ x ∈ M if the symbol σΦ(x, ξ): Ex → Fx is an isomorphism for all nonzero ξ ∈ Tx M. Φ0 ∞ Φ1 ∞ Φ2 ∞ Φ3 Φn ∞ Φn+1 A complex 0 −→ C (E0) −→ C (E1) −→ C (E2) −→ ... −→ C (En) −→ 0 is σ σΦi Φi+1 said to be elliptic at Ei if the symbol sequence Ei−1 −→ Ei −→ Ei+1 is exact for all ∗ ξ ∈ Tx M and for all x ∈ M. The link between these two forms of ellipticity is as follows. ∗ If we have a metric on each Ei then we can define a formal adjoint Φi : Ei → Ei−1. Linear algebra reveals that the complex is elliptic at Ei if and only if the Laplacian ∗ ∗ Φi Φi + Φi−1Φi−1 is an elliptic operator. One important implication of this is that an elliptic complex on a compact manifold has finite-dimensional cohomology groups [W, Theorem 5.2, p. 147]. Whether a complex yields interesting cohomological information is in this way directly related to whether or not the complex is elliptic. The following Proposition answers this question for the real Dolbeault complex.

Proposition 3.1.1 For p > 0, the upward complex

0 −→ Ωp,p −→ Ωp+1,p ⊕ Ωp,p+1 −→ Ωp+2,p ⊕ Ωp,p+2 −→ ... is elliptic everywhere except at the ﬁrst two spaces Ωp,p and Ωp+1,p ⊕ Ωp,p+1. For p = 0, the ‘leading edge’ complex

0 −→ Ω0,0 −→ Ω1,0 ⊕ Ω0,1 −→ Ω2,0 ⊕ Ω0,2 −→ ... is elliptic everywhere except at Ω1,0 ⊕ Ω0,1 = Ω1(M).

Proof. When p > q + 1, we have short sequences of the form

Ωp−1,q −→∂ Ωp,q −→∂ Ωp+1,q LLL (3.2) Ωq,p−1 −→∂ Ωq,p −→∂ Ωq,p+1.

Each such sequence is (a real subspace of) the direct sum of two elliptic sequences, and so is elliptic. Thus we have ellipticity at Ωp,q ⊕ Ωq,p whenever p ≥ q + 2.

27 This leaves us to consider the case when p = q, and the sequence

∂ Ωp+1,p −→∂ Ωp+2,2 −→ ... etc. p,p % LL 0 −→ Ω & (3.3) ∂ ∂ Ωp,p+1 −→ Ω2,p+2 −→ ... etc.

This fails to be elliptic. An easy and instructive way to see this is to consider the simplest 2 4-dimensional example M = C . 0 1 0 2 3 2 ∗ 2 ∼ 2 ab... Let e , e = I(e ), e and e = I(e ) form a basis for Tx C = C , and let e denote ea ∧ eb ∧ ... etc. Then I(e01) = e00 − e11 = 0, so e01 ∈ Λ1,1. The map from Λ1,1 2,1 1,2 0 01 01 0 to Λ ⊕ Λ is just the exterior diﬀerential d. Since σd(x, e )(e ) = e ∧ e = 0 the 1,1 2,1 1,2 symbol map σd :Λ → Λ ⊕ Λ is not injective, so the symbol sequence is not exact at Λ1,1. 123 2,1 1,2 0 123 Consider also e ∈ Λ ⊕ Λ . Then σ∂⊕∂(x, e )(e ) = 0, since there is no bundle 3,1 1,3 123 0 0 Λ ⊕ Λ . But e has no e -factor, so is not the image under σd(x, e ) of any form α ∈ Λ1,1. Thus the symbol sequence fails to be exact at Λ2,1 ⊕ Λ1,2. It is a simple matter to extend these counterexamples to higher dimensions and higher exterior powers. For k = 0, the situation is diﬀerent. It is easy to show that the complex

0 −→ C∞(M) −→d Ω1,0 ⊕ Ω0,1 −→∂⊕∂ Ω2,0 ⊕ Ω0,2 −→ ... etc. is elliptic everywhere except at (Ω1,0 ⊕ Ω0,1). This last sequence is given particular attention by Reyes-Carión[R, Lemma 2]. He shows that, when M is Kähler,ellipticity can be regained by adding the space hωi to the bundle Λ2,0 ⊕ Λ0,2, where ω is the real Kähler(1, 1)-form. The Real Dolbeault complex is thus elliptic except at the bottom of the isosceles triangle of spaces. Here the projection from d(Ωp,p) to Ωp+1,p ⊕ Ωp,p+1 is the identity, and arguments based upon non-trivial projection maps no longer apply. We shall see that this situation is closely akin to that of differential forms on quaternionic manifolds, and that techniques motivated by this example yield similar results.

3.2 Construction of the Double Complex

4n ∗ ∼ Let M be a quaternionic manifold. Then Tx M = V1 ⊗ E as an Sp(1)GL(n, H)- representation for all x ∈ M. Suppose we consider just the action of the Sp(1)-factor. 2n ∼ Then the (complexiﬁed) cotangent space eﬀectively takes the form V1 ⊗ C = 2nV1. k ∗ k Thus the Sp(1)-action on Λ T M is given by the representation Λ (2nV1). To work out the irreducible decomposition of this representation we compute the k 2 weight space decomposition of Λ (2nV1) from that of 2nV1. With respect to the action of a particular subgroup U(1)q ⊂ Sp(1), the representation 2nV1 has weights k +1 and −1, each occuring with multiplicity 2n. The weights of Λ (2nV1) are the k- k wise distinct sums of these. Each weight r in Λ (2nV1) must therefore be a sum of p occurences of the weight ‘+1’ and p − r occurences of the weight ‘−1’, where 2p − r = k

2This process for calculating the weights of tensor, symmetric and exterior powers is a standard technique in representation theory — see for example [FH, §11.2].

28 and 0 ≤ p ≤ k (from which it follows immediately that −k ≤ r ≤ k and r ≡ k mod 2). 2n The number of ways to choose the p ‘+1’ weights is p , and the number of ways 2n to choose the (p − r)‘−1’ weights is p−r , so the multiplicity of the weight r in the k representation Λ (2nV1) is

2n 2n Mult(r) = k+r k−r . 2 2 For r ≥ 0, consider the diﬀerence Mult(r)−Mult(r+2). This is the number of weight spaces of weight r which do not have any corresponding weight space of weight r + 2. Each such weight space must therefore be the highest weight space in an irreducible k ∗ subrepresentation Vr ⊆ Λ T M, from which it follows that the number of irreducibles k Vr in Λ (2nV1) is equal to Mult(r)−Mult(r + 2). This demonstrates the following Proposition:

Proposition 3.2.1 Let M 4n be a hypercomplex manifold. The decomposition into irreducibles of the induced representation of Sp(1) on ΛkT ∗M is

k k ∗ ∼ M 2n 2n 2n 2n Λ T M = k+r k−r − k+r+2 k−r−2 Vr, r=0 2 2 2 2 where r ≡ k mod 2.

p We will not always write the condition r ≡ k mod 2, assuming that q = 0 if q 6∈ Z. Definition 3.2.2 Let M 4n be a quaternionic manifold. Define the coefficient n 2n 2n 2n 2n k,r = k+r k−r − k+r+2 k−r−2 , 2 2 2 2

n and let Ek,r be the vector bundle associated to the Sp(1)-representation k,rVr. With this notation Proposition 3.2.1 is written

k k k ∗ ∼ M n M Λ T M = k,rVr = Ek,r. r=0 r=0

Our most important result is that this decomposition gives rise to a double complex of diﬀerential forms and operators on a quaternionic manifold.

∞ ∞ Theorem 3.2.3 The exterior derivative d maps C (M,Ek,r) to C (M,Ek+1,r+1 ⊕ Ek+1,r−1).

Proof. Let ∇ be a torsion-free linear connection on M preserving the quaternionic ∞ ∞ ∗ structure. Then ∇ : C (M,Ek,r) → C (M,Ek,r ⊗ T M). As Sp(1)-representations, this is ∞ n ∞ n ∇ : C (M, k,rVr) → C (M, k,rVr ⊗ 2nV1).

29 n ∼ n Using the Clebsch-Gordon formula we have k,rVr ⊗2nV1 = 2nk,r(Vr+1 ⊕Vr−1). Thus the k ∗ ∗ image of Ek,r under ∇ is contained in the Vr+1 and Vr−1 summands of Λ T M ⊗ T M. Since ∇ is torsion-free, d = ∧ ◦ ∇, so d maps (sections of) Ek,r to the Vr+1 and Vr−1 k+1 ∗ ∞ ∞ summands of Λ T M. Thus d : C (M,Ek,r) → C (M,Ek+1,r+1 ⊕ Ek+1,r−1). This allows us to split the exterior differential d into two ‘quaternionic Dolbeault operators’. k ∗ Definition 3.2.4 Let πk,r be the natural projection from Λ T M onto Ek,r. Define the operators

D : C∞(E ) → C∞(E ) D : C∞(E ) → C∞(E ) k,r k+1,r+1 and k,r k+1,r−1 (3.4) D = πk+1,r+1 ◦ d D = πk+1,r−1 ◦ d.

Theorem 3.2.3 is equivalent to the following:

Proposition 3.2.5 On a quaternionic manifold M, we have d = D + D, and so

2 D2 = DD + DD = D = 0.

Proof. The ﬁrst equation is equivalent to Theorem 3.2.3. The rest follows immediately from decomposing the equation d2 = 0.

Figure 3.2: The Double Complex

∞ C (E3,3) ...... etc. ¨¨* H D¨ HH ¨¨ HH ¨ D Hj ∞ C (E2,2) ¨* H ¨* D ¨ H D ¨ ¨¨ HH ¨¨ ¨¨ D HHj ¨¨ ∞ ∗ ∞ C (E1,1 = T M) C (E3,1) ...... etc. ¨* H ¨* H D¨¨ HH D¨¨ HH ¨ H ¨ H ¨¨ D HHj ¨¨ D HHj ∞ ∞ C (E0,0 = R) C (E2,0)

Here is our quaternionic analogue of the Dolbeault complex. There are strong sim- ilarities between this and the Real Dolbeault complex (Figure 3.1). Again, instead of a diamond as in the Dolbeault complex, the quaternionic version only extends upwards to form an isosceles triangle. This is essentially because there is one irreducible U(1)- representation for each integer, whereas there is one irreducible Sp(1)-representation only for each nonnegative integer. Note that this is a decomposition of real as well as complex diﬀerential forms; the operators D and D map real forms to real forms.

30 k By deﬁnition, the bundle Ek,k is the bundle A of (2.9) — they are both the subbundle k ∗ of Λ T M which includes all Sp(1)-representations of the form Vk. Thus the leading edge of the double complex

∞ D ∞ D ∞ D D ∞ D 0 −→ C (E0,0) −→ C (E1,1) −→ C (E2,2) −→ ... −→ C (E2n,2n) −→ 0 is precisely the complex (2.11) discovered by Salamon. Example 3.2.6 Four Dimensions This double complex is already very well-known and understood in four dimensions. 2 ∗ ∼ Here there is a splitting only in the middle dimension, Λ T M = V2 ⊕ 3V0. Let I, J 0 ∗ 1 0 and K be local almost complex structures at x ∈ M, and let e ∈ Tx M. Let e = I(e ), 2 0 3 0 0 3 ∗ ∼ e = J(e ) and e = K(e ). In this way we obtain a basis {e , . . . , e } for Tx M = H. Using the notation eijk... = ei ∧ ej ∧ ek ∧ ...etc., deﬁne the 2-forms

± 01 23 ± 02 31 ± 03 12 ω1 = e ± e , ω2 = e ± e , ω3 = e ± e . (3.5)

3 − − − − Then I, J and K all act trivially on the ωj , so E2,0 = hω1 , ω2 , ω3 i. The action of + sp(1) on the ωj is given by the multiplication table

+ + + + + I(ω1 ) = 0 I(ω2 ) = 2ω3 I(ω3 ) = −2ω3 + + + + + J(ω1 ) = −2ω3 J(ω2 ) = 0 J(ω3 ) = 2ω1 (3.6) + + + + + K(ω1 ) = 2ω2 K(ω2 ) = −2ω1 K(ω3 ) = 0.

These are the relations of the irreducible sp(1)-representation V2, and we see that E2,2 = + + + hω1 , ω2 , ω3 i. These bundles will be familiar to most readers: E2,2 is the bundle of self-dual 2- 2 2 forms Λ+ and E2,0 is the bundle of anti-self-dual 2-forms Λ−. The celebrated splitting 2 ∗ ∼ 2 2 Λ T M = Λ+ ⊕ Λ− is an invariant of the conformal class of any Riemannian 4-manifold, and I2 + J 2 + K2 = −4(∗ + 1), where ∗ :ΛkT ∗M → Λ4−kT ∗M is the Hodge star map. This also serves to illustrate why in four dimensions we make the deﬁnition that a quaternionic manifold is a self-dual conformal manifold. The relationship between quaternionic, almost complex and Riemannian structures in four dimensions is described in more detail in [S4, Chapter 7], a classic reference being [AHS]. Because there is no suitable quaternionic version of holomorphic coordinates, there is ∞ no ‘nice’ co-ordinate expression for a typical section of C (Ek,r). In order to determine the decomposition of a diﬀerential form, the simplest way the author has found is to use 2 2 2 the Casimir operator C = I + J + K . Consider a k-form α. Then α ∈ Ek,r if and only if (I2 + J 2 + K2)(α) = −r(r + 2)α. This mechanism also allows us to work out expressions for D and D acting on α.

∞ Lemma 3.2.7 Let α ∈ C (Ek,r). Then 1 1 Dα = − (r − 1) + (I2 + J 2 + K2) dα 4 r + 1

3We are assuming throughout that uppercase operators like I, J and K are acting as elements of a Lie algebra, not a Lie group. This makes no diﬀerence on T ∗M but is important on the exterior powers ΛkT ∗M. In particular, for k 6= 1 we no longer expect I2 = J 2 = K2 = −1, and by ‘act trivially’ we mean ‘annihilate’.

31 and 1 1 Dα = (r + 3) + (I2 + J 2 + K2) dα. 4 r + 1

Proof. We have dα = Dα + Dα, where Dα ∈ Ek+1,r+1 and Dα ∈ Ek+1,r−1. Applying the Casimir operator gives

(I2 + J 2 + K2)(dα) = −(r + 1)(r + 3)Dα − (r + 1)(r − 1)Dα.

Rearranging these equations gives Dα and Dα. ∞ ∞ Writing Dk,r for the particular map D : C (Ek,r) → C (Ek+1,r+1), we deﬁne the quaternionic cohomology groups

k,r Ker(Dk,r) HD (M) = . (3.7) Im(Dk−1,r−1) 3.3 Ellipticity and the Double Complex

In this section we shall determine where our double complex is elliptic and where it is not. It turns out that we have ellipticity everywhere except on the bottom two rows of the complex. This is exactly like the real Dolbeault complex of Figure 3.1, and though it is more diﬃcult to prove for the quaternionic version, the guiding principles which determine where the two double complexes are elliptic are very much the same in both cases. Here is the main result of this section:

Theorem 3.3.1 For 2k ≥ 4, the complex

D D D D D 0 → E2k,0 → E2k+1,1 → E2k+2,2 → ... → E2n+k,2n−k → 0 is elliptic everywhere except at E2k,0 and E2k+1,1, where it is not elliptic. For k = 1 the complex is elliptic everywhere except at E3,1, where it is not elliptic. For k = 0 the complex is elliptic everywhere.

The rest of this section provides a proof of this Theorem. Note the strong similarity between this Theorem and Proposition 3.1.1, the analogous result for the Real Dolbeault complex. Again, it is the isosceles triangle as opposed to diamond shape which causes ellipticity to fail for the bottom row, because d = D on E2k,0 and the projection from ∞ ∞ d(C (E2k,0)) to C (E2k+1,1) is the identity. On a complex manifold M 2n with holomorphic coordinates zj, the exterior forms dza1 ∧ ... ∧ dzap ∧ dz¯b1 ∧ ... ∧ dz¯bq span Λp,q. This allows us to decompose any form ω ∈ Λp,q, making it much easier to write down the kernels and images of maps which involve exterior multiplication. On a quaternionic manifold M 4n there is unfortunately no easy way to write down a local frame for the bundle Ek,r, because there is no quaternionic version of ‘holomorphic coordinates’. However, we can decompose Ek,r just enough to enable us to prove Theorem 3.3.1.

32 n A principal observation is that since ellipticity is a local property, we can work on H n without loss of generality. Secondly, since GL(n, H) acts transitively on H \{0}, if the σ σ σ e0 e0 e0 0 ∗ n symbol sequence ... −→ Ek,r −→ Ek+1,r+1 −→ ... is exact for any nonzero e ∈ T H n then it is exact for all nonzero ξ ∈ T ∗H . To prove Theorem 3.3.1, we choose one such 0 e and analyse the spaces Ek,r accordingly.

3.3.1 Decomposition of the Spaces Ek,r 0 ∗ n ∼ n n Let e ∈ Tx H = H and let (I, J, K) be the standard hypercomplex structure on H . As 1 0 2 0 3 0 0 3 ∼ in Example 3.2.6, define e = I(e ), e = J(e ) and e = K(e ), so that he , . . . , e i = H. In this way we single out a particular copy of H which we call H0, obtaining a (nonnatural) ∗ n ∼ n−1 splitting Tx H = H ⊕ H0 which is preserved by action of the hypercomplex structure. k n ∼ L4 k−l n−1 l This induces the decomposition Λ H = l=0 Λ H ⊗ Λ H0, which decomposes each k n Ek,r ⊂ Λ H according to how many differentials in the H0-direction are present. l Definition 3.3.2 Define the space Ek,r to be the subspace of Ek,r consisting of exterior forms with precisely l differentials in the H0-direction, i.e.

l k−l n−1 l Ek,r ≡ Ek,r ∩ (Λ H ⊗ Λ H0).

l k n Then Ek,r is preserved by the induced action of the hypercomplex structure on Λ H . 0 1 2 3 4 Thus we obtain an invariant decomposition Ek,r = Ek,r ⊕ Ek,r ⊕ Ek,r ⊕ Ek,r ⊕ Ek,r. Note 0 n n−1 that we can identify Ek,r on H with Ek,r on H . (Throughout the rest of this section, juxtaposition of exterior forms will denote exterior multiplication, for example αeij means α ∧ eij.) We can decompose these summands still further. Consider, for example, the bun- 1 1 0 1 2 3 dle Ek,r. An exterior form α ∈ Ek,r is of the form α0e + α1e + α2e + α3e , where k−1 n−1 k−1 n−1 ∼ αj ∈ Λ H . Thus α is an element of Λ H ⊗ 2V1, since H0 = 2V1 as an sp(1)- representation. Since α is in a copy of the representation Vr, it follows from the isomor- ∼ phism Vr ⊗ V1 = Vr+1 ⊕ Vr−1 that the αj must be in a combination of Vr+1 and Vr−1 0 0 representations, i.e. αj ∈ Ek−1,r+1 ⊕ Ek−1,r−1. We write

+ − + − 0 + − 1 + − 2 + − 3 α = α + α = (α0 + α0 )e + (α1 + α1 )e + (α2 + α2 )e + (α3 + α3 )e ,

+ 0 − 0 where αj ∈ Ek−1,r+1 and αj ∈ Ek−1,r−1. The following Lemma allows us to consider α+ and α− separately.

+ − 1 + − 1 Lemma 3.3.3 If α = α + α ∈ Ek,r then both α and α must be in Ek,r.

Proof. In terms of representations, the situation is of the form ∼ (pVr+1 ⊕ qVr−1) ⊗ 2V1 = 2p(Vr+2 ⊕ Vr) ⊕ 2q(Vr ⊕ Vr−2),

+ − where α ∈ pVr+1 and α ∈ qVr−1. For α to be in the representation 2(p + q)Vr, its components in the representations 2pVr+2 and 2qVr−2 must both vanish separately. + The component in 2pVr+2 comes entirely from α , so for this to vanish we must have

33 + − α ∈ 2(p + q)Vr independently of α . Similarly, for the component in 2qVr−2 to vanish, − we must have α ∈ 2(p + q)Vr. 1 0 Thus we decompose the space Ek,r into two summands, one coming from Ek−1,r−1 ⊗ 0 2V1 and one from Ek−1,r+1 ⊗2V1. We extend this decomposition to the cases l = 0, 2, 3, 4, defining the following notation. l,m l Definition 3.3.4 Define the bundle Ek,r to be the subbundle of Ek,r arising from k−l n−1 Vm-type representations in Λ H . In other words,

l,m 0 l Ek,r ≡ (Ek−l,m ⊗ Λ H0) ∩ Ek,r.

l,m To recapitulate: for the space Ek,r , the bottom left index k refers to the exterior n power of the form α ∈ ΛkH ; the bottom right index r refers to the irreducible Sp(1)- representation in which α lies; the top left index l refers to the number of differentials in the H0-direction and the top right index m refers to the irreducible Sp(1)-representation k−a n−1 of the contributions from Λ H before wedging with forms in the H0-direction. This may appear slightly fiddly: it becomes rather simpler when we consider the specific splittings which Definition 3.3.4 allows us to write down.

l,m Lemma 3.3.5 Let Ek,r be as above. We have the following decompositions:

0 0,r 1 1,r+1 1,r−1 2 2,r+2 2,r 2,r−2 Ek,r = Ek,r Ek,r = Ek,r ⊕ Ek,r Ek,r = Ek,r ⊕ Ek,r ⊕ Ek,r

3 3,r+1 3,r−1 4 4,r Ek,r = Ek,r ⊕ Ek,r and Ek,r = Ek,r .

Proof. The ﬁrst isomorphism is trivial, as is the last (since the hypercomplex structure 4 acts trivially on Λ H0). The second isomorphism is Lemma 3.3.3, and the fourth follows 3 ∼ in exactly the same way since Λ H0 = 2V1 also. The middle isomorphism follows a similar argument. 2,r Recall the self-dual forms and anti-self-dual forms in Example 3.2.6. The bundle Ek,r 2 splits according to whether its contribution from Λ H0 is self-dual or anti-self-dual. We 2,r+ 2,r− 2,r 2,r+ 2,r− will call these summands Ek,r and Ek,r respectively, so Ek,r = Ek,r ⊕ Ek,r .

3.3.2 Lie in conditions

We have analysed the bundle Ek,r into a number of diﬀerent subbundles. We now determine when a particular exterior form lies in one of these subbundles. Consider s1...sa t1...ta 0 a form α = α1e + α2e + ... etc. where αj ∈ Ek−a,b. For α to lie in one of the a,b spaces Ek,r the αj will usually have to satisfy some simultaneous equations. Since these are the conditions for a form to lie in a particular Lie algebra representation, we will refer to such equations as ‘Lie in conditions’. 0 0,r To begin with, we mention three trivial Lie in conditions. Let α ∈ Ek,r. That α ∈ Ek,r 0123 4,r 0123 is obvious, as is αe ∈ Ek,r , since wedging with e has no eﬀect on the sp(1)-action.

34 − 01 23 − 02 31 Likewise, the sp(1)-action on the anti-self-dual 2-forms ω1 = e − e , ω2 = e − e − 03 12 − 2,r− and ω3 = e − e is trivial, so αωj ∈ Ek,r for all j = 1, 2, 3. This leaves the following three situations: those arising from taking exterior products + with 1-forms, 3-forms and the self-dual 2-forms ωj . As usual when we want to know which representation an exterior form is in, we apply the Casimir operator.

The cases l = 1 and l = 3

0 0 1 2 3 1,r 1,r Let αj ∈ Ek,r. Then α = α0e + α1e + α2e + α3e ∈ Ek+1,r+1 ⊕ Ek+1,r−1, and α is 1,r 2 2 2 entirely in Ek+1,r+1 if and only if (I + J + K )α = −(r + 1)(r + 3)α. By the usual (Leibniz) rule for a Lie algebra action on a tensor product, we have that 2 j 2 j j 2 j I (αje ) = I (αj)e + 2I(αj)I(e ) + αjI (e ), etc. Thus

3 2 2 2 X h 2 2 2 j 2 2 2 j (I + J + K )α = (I + J + K )(αj)e + αj(I + J + K )(e ) + j=0 j j j i + 2 I(αj)I(e ) + J(αj)J(e ) + K(αj)K(e )

3 X j j j = −r(r + 2)α − 3α + 2 I(αj)I(e ) + J(αj)J(e ) + K(αj)K(e ) j=0

2 1 0 3 2 = (−r − 2r − 3)α + 2 I(α0)e − I(α1)e + I(α2)e − I(α3)e + 2 3 0 1 +J(α0)e − J(α1)e − J(α2)e + J(α3)e + 3 2 1 0 +K(α0)e + K(α1)e − K(α2)e − K(α3)e . (3.8)

1,r For α ∈ Ek+1,r+1 we need this to be equal to −(r + 1)(r + 3)α, which is the case if and only if

1 0 3 2 2 3 0 1 −rα = I(α0)e − I(α1)e + I(α2)e − I(α3)e + J(α0)e − J(α1)e − J(α2)e + J(α3)e + 3 2 1 0 + K(α0)e + K(α1)e − K(α2)e − K(α3)e .

j Since the αj have no e -factors and the action of I, J and K preserves this property, this equation can only be satisﬁed if it holds for each of the ej-components separately. 1,r We conclude that α ∈ Ek+1,r+1 if and only if α0, α1, α2 and α3 satisfy the following Lie in conditions: 4

rα0 − I(α1) − J(α2) − K(α3) = 0 rα + I(α ) + J(α ) − K(α ) = 0 1 0 3 2 (3.9) rα2 − I(α3) + J(α0) + K(α1) = 0 rα3 + I(α2) − J(α1) + K(α0) = 0. 1,r 2 2 2 Suppose instead that α ∈ Ek+1,r−1. Then (I +J +K )α = −(r−1)(r+1)α. Putting 1,r this alternative into Equation (3.8) gives the result that α ∈ Ek+1,r−1 if and only if

4Our interest in these conditions arises from a consideration of exterior forms, but the equations describe sp(1)-representations in general: they are the conditions that α ∈ Vr ⊗ V1 must satisfy to be in ∼ the Vr+1 subspace of Vr+1 ⊕ Vr−1 = Vr ⊗ V1. The other Lie in conditions have similar interpretations.

35 (r + 2)α0 + I(α1) + J(α2) + K(α3) = 0 (r + 2)α − I(α ) − J(α ) + K(α ) = 0 1 0 3 2 (3.10) (r + 2)α2 + I(α3) − J(α0) − K(α1) = 0 (r + 2)α3 − I(α2) + J(α1) − K(α0) = 0. 123 032 013 021 3,r 3,r Consider now α = α0e + α1e + α2e + α3e ∈ Ek+3,r+1 ⊕ Ek+3,r−1. Since 3 ∼ 3,r Λ H0 = H0, the Lie in conditions are exactly the same: for α to be in Ek+3,r+1 we need 3,r the αj to satisfy Equations (3.9), and for α to be in Ek+3,r−1 we need the αj to satisfy Equations (3.10).

The case l = 2

0 − We have already noted that wedging a form β ∈ Ek,r with an anti-self-dual 2-form ωj − 2,r− has no eﬀect on the sp(1)-action, so βωj ∈ Ek+2,r. Thus we only have to consider the + + + ∼ 2 eﬀect of wedging with the self-dual 2-forms hω1 , ω2 , ω3 i = V2 ⊂ Λ H0. By the Clebsch- ∼ Gordon formula, the decomposition takes the form Vr ⊗ V2 = Vr+2 ⊕ Vr ⊕ Vr−2. Thus + + + for β = β1ω1 + β2ω2 + β3ω3 we want to establish the Lie in conditions for β to be in 2,r 2,r+ 2,r Ek+2,r+2, Ek+2,r and Ek+2,r−2. We calculate these Lie in conditions in a similar fashion to the previous cases, by considering the action of the Casimir operator I2 +J 2 +K2 on β and using the multiplication table (3.6). The following Lie in conditions are then easy to deduce:  (r + 4)β1 = J(β3) − K(β2) 2,r  β ∈ Ek+2,r+2 ⇐⇒ (r + 4)β2 = K(β1) − I(β3) (3.11)  (r + 4)β3 = I(β2) − J(β1).  2β1 = J(β3) − K(β2) 2,r+  β ∈ Ek+2,r ⇐⇒ 2β2 = K(β1) − I(β3) (3.12)  2β3 = I(β2) − J(β1).  (2 − r)β1 = J(β3) − K(β2) 2,r  β ∈ Ek+2,r−2 ⇐⇒ (2 − r)β2 = K(β1) − I(β3) (3.13)  (2 − r)β3 = I(β2) − J(β1). Equation 3.12 is particularly interesting. Since this equation singles out the Vr-representation in the direct sum Vr+2 ⊕ Vr ⊕ Vr+2, it must have dim Vr = r + 1 linearly independent solutions. Let β0 ∈ Vr and let β1 = I(β0), β2 = J(β0), β3 = K(β0). Using the Lie algebra relations 2I = [J, K] = JK − KJ, it is easy to see that β1, β2 and β3 satisfy Equation 3.12. Moreover, there are r + 1 linearly independent solutions of this form (for r 6= 0). We conclude that all the solutions of Equation (3.12) take the form β1 = I(β0), β2 = J(β0), β3 = K(β0).

3.3.3 The Symbol Sequence and Proof of Theorem 3.3.1 We now describe the principal symbol of D, and examine its behaviour in the context of the decompositions of Deﬁnition 3.3.2 and Lemma 3.3.5. This leads to a proof of Theorem 3.3.1. First we obtain the principal symbol from the formula for D in Lemma 3.2.7 by replacing dα with αe0.

36 n 0 ∗ n Proposition 3.3.6 Let x ∈ H , e ∈ Tx H and α ∈ Ek,r. The principal symbol 0 mapping σD(x, e ): Ek,r → Ek+1,r+1 is given by 1 σ (x, e0)(α) = (r + 2)αe0 − I(α)e1 − J(α)e2 − K(α)e3 . D 2(r + 1)

Proof. Replacing dα with αe0 in the formula for D obtained in Lemma 3.2.7, we have 1 1 σ (x, e0)(α) = − (r − 1) + (I2 + J 2 + K2) αe0 D 4 r + 1 −1 h = ((r − 1)(r + 1) − r(r + 2) − 3 ) αe0 4(r + 1) i +2 I(α)e1 + J(α)e2 + K(α)e3 1 = (r + 2)αe0 − I(α)e1 − J(α)e2 − K(α)e3 , 2(r + 1) as required.

0 l,m Corollary 3.3.7 The principal symbol σD(x, e ) maps the space Ek,r to the space l+1,m Ek+1,r+1.

Proof. We already know that σD : Ek,r → Ek+1,r+1, by definition. Using Lemma 3.3.6, 0 we see that σD(x, e ) increases the number of differentials in the H0-direction by one, so the index l increases by one. The only action in the other directions is the sp(1)-action, n−1 which preserves the irreducible decomposition of the contribution from Λk−aH , so the index m remains the same. 0 (To save space we shall use σ as an abbreviation for σD(x, e ) for the rest of this section.) l The point of all this work on decomposition now becomes apparent. Since σ : Ek,r → l+1 Ek+1,r+1, we can reduce the (somewhat indefinite) symbol sequence

σ σ σ σ ... −→ Ek−1,r−1 −→ Ek,r −→ Ek+1,r+1 −→ ... etc. to the 5-space sequence

σ 0 σ 1 σ 2 σ 3 σ 4 σ 0 −→ Ek−2,r−2 −→ Ek−1,r−1 −→ Ek,r −→ Ek+1,r+1 −→ Ek+2,r+2 −→ 0. (3.14) Using Lemma 3.3.5 as well, we can analyse this sequence still further according to the diﬀerent (top right) m-indices, obtaining three short sequences (for k ≥ 2, k ≡ r mod 2)

2,r+2 3,r+2 4,r+2 0 → Ek,r → Ek+1,r+1 → Ek+2,r+2 → 0 ⊕ ⊕ 1,r 2,r 3,r 0 → Ek−1,r−1 → Ek,r → Ek+1,r+1 → 0 ⊕ ⊕ 0,r−2 1,r−2 2,r−2 0 → Ek−2,r−2 → Ek−1,r−1 → Ek,r → 0. (3.15)

37 This reduces the problem of determining where the operator D is elliptic to the problem of determining when these three sequences are exact. For a sequence 0 → A → B → C → 0 to be exact, it is necessary that dim A − dim B + dim C = 0. Given this condition, if the sequence is exact at any two out of A, B and C it is exact at the third. We shall show that for r 6= 0 this dimension sum does equal zero.

Lemma 3.3.8 For r > 0, each of the sequences in (3.15) satisﬁes the dimension condition above, i.e. the alternating sum of the dimensions vanishes.

l,m Proof. Let r > 0. We calculate the dimensions of the spaces Ek,r for l = 0,..., 4. Recall n 0 n−1 the notation Ek,r = k,rVr from Deﬁnition 3.2.2. It is clear that dim Ek,r = (r + 1)k,r , 0 n n−1 0,r−2 n−1 since Ek,r on H is simply Ek,r on H . Thus dim Ek−2,r−2 = (r − 1)k−2,r−2 and 4,r+2 n−1 dim Ek+2,r+2 = (r + 3)k−2,r+2. 0 The cases a = 1 and a = 3 are easy to work out since they are of the form Ek,r ⊗ 2V1. 1,r−2 n−1 1,r n−1 For a = 1, we have dim Ek−1,r−1 = 2rk−2,r−2 and dim Ek−1,r−1 = 2rk−2,r. For a = 3, 3,r n−1 3,r+2 n−1 dim Ek+1,r+1 = 2(r + 2)k−2,r and dim Ek+1,r+1 = 2(r + 2)k−2,r+2. The case a = 2 is slightly more complicated, as we have to take into account exterior 2 products with the self-dual 2-forms V2 and anti-self-dual 2-forms 3V0 in Λ H0. The spaces 2,r+2 2,r−2 Ek,r and Ek,r receive contributions only from the self-dual part V2, from which we 2,r+2 n−1 2,r−2 n−1 infer that dim Ek,r = (r + 1)k−2,r+2 and dim Ek,r = (r + 1)k−2,r−2. Finally, the 2,r+ n−1 2,r− n−1 space Ek,r has dimension (r+1)k−2,r and the space Ek,r has dimension 3(r+1)k−2,r, 2,r n−1 giving Ek,r a total dimension of 4(r + 1)k−2,r. It is now a simple matter to verify that for the top sequence of (3.15)

n−1 k−2,r+2(r + 1 − 2(r + 2) + r + 3) = 0, for the middle sequence

n−1 k−2,r(2r − 4(r + 1) + 2(r + 2)) = 0, and for the bottom sequence

n−1 k−2,r−2(r − 1 − 2r + r + 1) = 0.

The case r = 0 is diﬀerent. Here the bottom sequence of (3.15) disappears altogether, the top sequence still being exact. Exactness is lost in the middle sequence. Since the n−1 ∼ n−1 isomorphism k−2,0V0 ⊗V2 = k−2,0V2 gives no trivial V0-representations, there is no space 2,0+ 2,0 Ek,0 . Thus Ek,0 is ‘too small’ — we are left with a sequence

n−1 n−1 0 −→ 3k−2,0V0 −→ 2k−2,0V1 −→ 0,

2 which cannot be exact. (As there is no space E0,0, this problem does not arise for the leading edge 0 → E0,0 → E1,1 → ... etc.) We are ﬁnally in a position to prove Theorem 3.3.1, which now follows from:

38 Proposition 3.3.9 When r 6= 0, the three sequences of (3.15) are exact.

σ 2,r+2 σ 3,r+2 σ 4,r+2 Proof. Consider ﬁrst the top sequence 0 −→ Ek,r −→ Ek+1,r+1 −→ Ek+2,r+2 −→ 0. 3,r+2 4,r+2 The Clebsch-Gordon formula shows that there are no spaces Ek+1,r−1 or Ek+2,r . Thus 2,r+2 3,r+2 D = 0 on Ek,r and Ek+1,r+1, so D = d for the top sequence. It is easy to check using 0 2,r+2 3,r+2 the relevant Lie in conditions that the map ∧e : Ek,r → Ek+1,r+1 is injective and the 0 3,r+2 4,r+2 map ∧e : Ek+1,r+1 → Ek+2,r+2 is surjective. 1 0 1 2 3 1 To show exactness at Ek−1,r−1, consider α = α0e + α1e + α2e + α3e ∈ Ek−1,r−1.A calculation using Proposition 3.3.6 shows that 1 σ(α) = (rα + I(α ))e10 + (rα + J(α ))e20 + (rα + K(α ))e30+ (3.16) 2r 1 0 2 0 3 0 32 13 21 + (2α1 − J(α3) + K(α2))e + (2α2 − K(α1) + I(α3))e + (2α3 − I(α2) + K(α1))e .

j Since the αi have no e -components, σ(α) = 0 if and only if all these components 1 1 1 vanish. This occurs if and only if α1 = − r I(α0), α2 = − r J(α0), α3 = − r K(α0) (since as remarked in Section 3.3.1 these equations also guarantee that 2α1 −J(α3)+K(α2) = 0 etc.), in which case it is clear that

2(r+1) σ(α) = 0 ⇐⇒ α = σ r α0 .

0 1 2 This shows that the sequence Ek−2,r−2 → Ek−1,r−1 → Ek,r is exact. Restricting to 1,r 1,r−2 Ek−1,r−1 and Ek−1,r−1, we see that exactness holds at these spaces in the middle and bottom sequences respectively of (3.15). 0 Consider α ∈ Ek−2,r−2. Then 1 σ(α) = rαe0 − I(α)e1 − J(α)e2 − K(α)e3 . 2(r − 1)

0,r−2 Since these are linearly independent, σ(α) = 0 if and only if α = 0, and σ : Ek−2,r−2 → 1,r−2 0,r−2 σ 1,r−2 σ Ek−1,r−1 is injective. Hence the bottom sequence 0 −→ Ek−2,r−2 −→ Ek−1,r−1 −→ 2,r−2 Ek,r −→ 0 is exact. 1,r σ 2,r σ 3,r Finally, we show that the middle sequence 0 −→ Ek−1,r−1 −→ Ek,r −→ Ek+1,r+1 −→ 2,r 0 is exact at Ek,r , which is now suﬃcient to show that the sequence is exact. + + + 2,r+ Let β = β1ω1 + β2ω2 + β3ω3 ∈ Ek,r . Recall the Lie in condition (3.12) that β 1 + + + 0 must take the form β = r I(β0)ω1 + J(β0)ω2 + K(β0)ω3 for some β0 ∈ Ek−2,r. (The 1 r -factor makes no diﬀerence here and is useful for cancellations.) Thus a general element 2,r of Ek,r is of the form 1 β + γ = I(β )ω+ + J(β )ω+ + K(β )ω+ + γ ω− + γ ω− + γ ω−, r 0 1 0 2 0 3 1 1 2 2 3 3 0,r for β0, γj ∈ Ek−2,r. A similar calculation to that of (3.16) shows that   (r + 2)β0 + I(γ1) + J(γ2) + K(γ3) = 0  (r + 2)γ − I(β ) − J(γ ) + K(γ ) = 0 σ(β + γ) = 0 ⇐⇒ 1 0 3 2 (r + 2)γ2 + I(γ3) − J(β0) − K(γ1) = 0   (r + 2)γ3 − I(γ2) + J(γ1) − K(β0) = 0.

39 0 1 2 3 But this is exactly the Lie in condition (3.10) which we need for β0e + γ1e + γ2e + γ3e 1,r to be in Ek−1,r−1, in which case we have

0 1 2 3 β + γ = σ 2(β0e + γ1e + γ2e + γ3e ) .

2,r This demonstrates exactness at Ek,r and so the middle sequence is exact. 0 As a counterexample for the case r = 0 and k ≥ 4, consider α ∈ Ek−4,0. Then 0123 4,0 0123 αe ∈ Ek,0 and σ(αe ) = 0, so σ : Ek,0 → Ek+1,1 is not injective, which is exactly the same as saying that the symbol sequence is not exact at Ek,0. It is easy to see that this counterexample does not arise when k = 0 or 2, and to show that the maps σ : E0,0 → E1,1 and σ : E2,0 → E3,1 are injective. 0 As a counterexample for the case r = 1 and k ≥ 2, consider α ∈ Ek−2,0. Then 123 3,0 123 0 4 123 123 0 αe ∈ Ek+1,1 and αe ∧ e ∈ Ek+2,0. Thus σ(αe ) = 0. Since αe has no e - 123 2 components at all it is clear that αe 6= σ(β) for any β ∈ Ek,0. Thus the symbol sequence fails to be exact at Ek+1,1. Again, it is easy to see that this counterexample σ σ does not arise when k = 0, and to show that the sequence E0,0 −→ E1,1 −→ E2,2 is exact at E1,1. This concludes our proof of Theorem 3.3.1.

3.4 Quaternion-valued forms on Hypercomplex Man- ifolds

Let M be a hypercomplex manifold. Then M has a triple (I, J, K) of complex structures which we can identify globally with the imaginary quaternions. Thus we have globally defined operators which generate the sp(1)-action on ΛkT ∗M. Consider also the quaternions themselves. Equation (2.6) describes the n Sp(1)GL(n, H)-representation on H as V1 ⊗ E. In the case n = 1 this reduces to the representation ∼ H = V1 ⊗ V1, (3.17) where we can interpret the left-hand copy of V1 as the left-action (p, q) 7→ pq, and the −1 right-hand copy of V1 as the right-action (p, q) 7→ qp , for q ∈ H and p ∈ Sp(1). We can now use our globally defined hypercomplex structure to combine the Sp(1)- actions on H and ΛkT ∗M. This motivates a thorough investigation of quaternion-valued forms on hypercomplex manifolds. Consider, for example, quaternion-valued exterior n forms in the bundle Ek,r = k,rVr. The Sp(1)-action on these forms is described by the representation ∼ n H ⊗ Ek,r = V1 ⊗ V1 ⊗ k,rVr. Leaving the left H-action untouched, we consider the effect of the right H-action and the hypercomplex structure simultaneously. This amounts to applying the operators

I : α → I(α) − αi1, J : α → J(α) − αi2 and K : α → K(α) − αi3

n to α ∈ H ⊗ Ek,r. Under this diagonal action the tensor product V1 ⊗ k,rVr splits, giving the representation ∼ n H ⊗ Ek,r = V1 ⊗ k,r(Vr+1 ⊕ Vr−1). (3.18)

40 Each of these summands inherits the structure of a left H-module from the left-action V1, which is not aﬀected by our splitting. This situation mirrors our discussion of real and complex forms on complex manifolds. There is a decomposition of real-valued forms, which is taken further when we consider complex-valued forms. In the same way, considering quaternion-valued forms on a hypercomplex manifold allows us to take our decomposition further. This point of view turns out to be very fruitful. It will, over the next few chapters, lead to quaternionic analogues of holomorphic functions and k-forms, the holomorphic tangent and cotangent spaces, and complex Lie groups and Lie algebras. The algebraic foundation for this geometry lies in considering objects like our left n H-modules in Equation (3.18). Each of the summands V1 ⊗ k,rVr±1 is a left H-module ∼ n n which arises as a submodule of H ⊗ Ek,r = (r + 1)k,rH . Thus each summand is an n H-linear submodule of H . In the next chapter we will introduce a new algebraic theory which is based upon such objects.

41 Chapter 4

Developments in Quaternionic Algebra

This chapter describes an algebraic theory which will be central to our description of hypercomplex geometry. The theory is that of my supervisor, Dominic Joyce, and is n presented in [J1]. The basic objects of study are H-submodules U of H ⊗ R . Joyce n shows that the inclusion ιU : U → H is determined up to isomorphism by the H- module structure of U and the choice of a real vector subspace U 0 ⊂ U satisfying a certain condition. The pair (U, U 0) is an augmented H-module, or AH-module. The most important discovery in [J1] is a canonical tensor product for AH-modules with interesting properties. (Recall from Section 1.3 that the most obvious definitions of a tensor product over the quaternions are not especially fruitful.) For two AH-modules m n mn U ⊂ H and V ⊂ H , we can define a unique AH-module U⊗HV ⊂ H . The operation ‘ ⊗H ’ will be called the quaternionic tensor product. It has similar properties to the tensor product over a commuting field; for example it is both associative and commutative. This allows us to develop the algebra of AH-modules as a parallel to that of vector spaces over R or C. This analogy is particularly strong for certain well-behaved AH-modules which will be called stable AH-modules. There are other algebraic operations which are equivalent to Joyce’s quaternionic tensor product. A sheaf-theoretic point of view is presented by Quillen [Q], in which he discovers a contravariant equivalence of tensor categories between AH-modules and regular 1 sheaves on a real form of CP . This allows us to classify all AH-modules and determine their tensor products. In the next chapter, we will see that the most important classes of AH-modules are conveniently described and manipulated using Sp(1)-representations.

4.1 The Quaternionic Algebra of Joyce

The following is a summary of parts of Joyce’s theory of quaternionic algebra. The interested reader should consult [J1] for more details and proofs.

4.1.1 AH-Modules We begin by deﬁning H-modules and their dual spaces. A (left) H-module is a real vector space U with an action of H on the left which we write as (q, u) 7→ q · u or qu, such that p(q(u)) = (pq)(u) for p, q ∈ H and u ∈ U. For our purposes, all H-modules will

42 be left H-modules. By dim U we will always mean the dimension of U as a real vector space, even if U is an H-module. We write U ∗ for the dual vector space of U. If U is an H-module we also define the dual H-module U × of linear maps α : U → H that satisfy α(qu) = qα(u) for all q ∈ H and u ∈ U. If q ∈ H and α ∈ U ×, define q · α by (q · α)(u) = α(u)q for u ∈ U. Then q · α ∈ U ×, and U × is a (left) H-module. Dual H-modules behave just like dual vector spaces. Definition 4.1.1 [J1, Definition 2.2] Let U be an H-module. Let U 0 be a real vector subspace of U. Define a real vector subspace U † of U × by

† × U = {α ∈ U : α(u) ∈ I for all u ∈ U 0}. (4.1) Conversely, U † determines U 0 (at least for ﬁnite-dimensional U) by

0 U = {u ∈ U : α(u) ∈ I for all α ∈ U †}. (4.2) An augmented H-module, or AH-module, is a pair (U, U 0) such that if u ∈ U and α(u) = 0 0 for all α ∈ U †, then u = 0. We consider H to be an AH-module, with H = I.AH- modules should be thought of as the quaternionic analogues of real vector spaces.

Usually we will refer to U itself as an AH-module, assuming that U 0 is also given. If we consider only the real part Re(α(u)) (for u ∈ U and α ∈ U ×), we can interpret U × as the dual of U as a real vector space, and then U † is the annihilator of U 0. Thus if U is finite-dimensional, dim U 0 + dim U † = dim U = dim U × and an isomorphism U ∼= U × determines an isomorphism U/U 0 ∼= U †. Let U be an AH-module and let u, v ∈ U such that α(u) = α(v) for all α ∈ U †. Then since α is a linear map we have α(u − v) = 0 for all α ∈ U † and it follows from Definition 4.1.1 that u = v. Thus U is an AH-module if and only if each u ∈ U is uniquely determined by the values of α(u) for α ∈ U †. In effect, the definition of an AH- module demands that U 0 should not be too large. Definition 4.1.1 demands for any u ∈ U, its H-linear span H · u should not be entirely contained in U 0, so dim(U 0 ∩ H · u) ≤ 3. If V is an AH-module, we say that U is an AH-submodule of V if U is an H-submodule of V and U 0 = U ∩V 0. As U † is the restriction of V † to U, if α(u) = 0 for all α ∈ U † then u = 0, so U is an AH-module. Define W to be the quotient H-module V/U and define W 0 to be the real subspace (V 0 + U)/U of W . We would like to define (W, W 0) to be the quotient AH-module V/U. However, there is a catch: W may not be an AH-module, as the condition in Definition 4.1.1 may not be satisfied. 3 Example 4.1.2 [J1, Definition 6.1] Let Y ⊂ H be the set Y = {(q1, q2, q3): q1i1 + ∼ 2 0 3 q2i2 + q3i3 = 0}. Then Y = H is a left H-module. Define a real subspace Y = Y ∩ I ; 0 so Y = {(q1, q2, q3): qj ∈ I and q1i1 + q2i2 + q3i3 = 0}. Then dim Y = 8 and dim Y 0 = 5. 0 Let ν : Y → H, ν(q1, q2, q3) = i1q1 + i2q2 + i3q3. Then im(ν) = I and ker(ν) = Y . 0 ∼ † ∗ † ∗ ∼ ∼ Since Y/Y = (Y ) , ν induces an isomorphism (Y ) = I = V2. Here is the natural concept of linear map between AH-modules: Definition 4.1.3 Let U, V be AH-modules and let φ : U → V be H-linear. We say that φ is an AH-morphism if φ(U 0) ⊂ V 0. If φ is also an isomorphism of H-modules we say φ is an AH-isomorphism.

43 One obvious question is whether we can classify AH-modules up to AH-isomorphism. Real and complex vector spaces are classiﬁed by dimension, but clearly this is not true 0 ∼ n for AH-modules, as there are several choices of U for each H-module U = H . We will return to this question in some detail later. We note the following points:

• If φ : U → V and ψ : V → W are AH-morphisms, then ψ ◦ φ : U → W is an AH-morphism.

• Let U, V be AH-modules and φ : U → V an AH-morphism. Deﬁne an H-linear map φ× : V × → U × by φ×(β)(u) = β(φ(u)) for β ∈ V × and u ∈ U. Then φ(U 0) ⊂ V 0 implies that φ×(V †) ⊂ U †.

• Let U be an AH-module. Then H⊗(U †)∗ is an H-module, with H-action p·(q⊗x) = † ∗ (pq) ⊗ x. Deﬁne a map ιU : U → H ⊗ (U ) by ιU (u) · α = α(u), for u ∈ U and † † ∗ α ∈ U . Then ιU is H-linear, so that ιU (U) is an H-submodule of H ⊗ (U ) . † Suppose u ∈ ker ιU . Then α(u) = 0 for all α ∈ U , so that u = 0 as U is an ∼ AH-module. Thus ιU is injective, and ιU (U) = U.

0 † ∗ • From Equation (4.2), it follows that ιU (U ) = ιU (U) ∩ (I ⊗ (U ) ). Thus the 0 AH-module (U, U ) is determined by the H-submodule ιU (U).

This shows as promised that every AH-module is isomorphic to a (left) submodule n n of (H ⊗ R , I ⊗ R ) for n = dim(U †)∗. Example 4.1.2 shows how ths works for the 2 † ∗ ∼ AH-module Y . As an abstract AH-module Y is isomorphic to H and (Y ) = V2. One of the easiest and most symmetrical ways to obtain Y is as an 8-dimensional subspace 3 of H = H ⊗ V2. We will ﬁnd this version of events very useful in many situations, as we shall see immediately.

4.1.2 The Quaternionic Tensor Product

Let U and V be AH-modules. Then they can be regarded as subspaces of H ⊗ (U †)∗ and H ⊗ (V †)∗ respectively. Since the H-action on both of these is the same, we can paste these AH-modules together to get a product AH-module. Here is the key idea of the theory: Definition 4.1.4 [J1, Definition 4.2] Let U, V be AH-modules. Then H ⊗ (U †)∗ ⊗ (V †)∗ is an H-module, with H-action p · (q ⊗ x ⊗ y) = (pq) ⊗ x ⊗ y. Exchanging the factors of † ∗ † ∗ † ∗ † ∗ H and (U ) , we may regard (U ) ⊗ ιV (V ) as a subspace of H ⊗ (U ) ⊗ (V ) . Thus † ∗ † ∗ † ∗ † ∗ ιU (U) ⊗ (V ) and (U ) ⊗ ιV (V ) are AH-submodules of H ⊗ (U ) ⊗ (V ) . We define their intersection to be the quaternionic tensor product of U and V ,

† ∗ † ∗ † ∗ † ∗ U⊗HV = (ιU (U) ⊗ (V ) ) ∩ ((U ) ⊗ ιV (V )) ⊂ H ⊗ (U ) ⊗ (V ) . (4.3)

0 0 † ∗ † ∗ The vector subspace (U⊗HV ) is then given by (U⊗HV ) = (U⊗HV )∩(I⊗(U ) ⊗(V ) ) and with this deﬁnition U⊗HV is an AH-module. The operation ⊗H will be called the quaternionic tensor product.

44 A few words of explanation may be useful at this point. At the end of Chapter 1 we m n ∼ mn saw that it is possible to define a sort of tensor product H ⊗HH = H , but that this is not really any different from taking tensor products over R. The theory of AH-modules and the quaternionic tensor product is a way of taking the quaternionic behaviour into account. How U 0 behaves in relation to the H-action determines a particular subspace n ιU (U) of H ⊗ R . The quaternionic tensor product is the natural way of combining m n these choices for AH-modules U ⊆ H ⊗ R and V ⊆ H ⊗ R into an AH-module mn U⊗HV ⊆ H ⊗ R . We also define the tensor product of two AH-morphisms: Definition 4.1.5 Let U, V, W, X be AH-modules, and let φ : U → W and ψ : V → X be AH-morphisms. Then φ×(W †) ⊂ U † and ψ×(X†) ⊂ V †. Taking the duals gives maps (φ×)∗ :(U †)∗ → (W †)∗ and (ψ×)∗ :(V †)∗ → (X†)∗. Combining these, we have a map

× ∗ × ∗ † ∗ † ∗ † ∗ † ∗ id ⊗(φ ) ⊗ (ψ ) : H ⊗ (U ) ⊗ (V ) → H ⊗ (W ) ⊗ (X ) . (4.4) × ∗ × ∗ Deﬁne φ⊗Hψ : U⊗HV → W ⊗HX to be the restriction of id ⊗(φ ) ⊗ (ψ ) to U⊗HV . Then φ⊗Hψ is an AH-morphism from U⊗HV to W ⊗HX . This is the quaternionic tensor product of φ and ψ.

It can be proved [J1, Lemma 4.3] that there are canonical AH-isomorphisms ∼ ∼ ∼ H⊗HU = U, U⊗HV = V ⊗HU and (U⊗HV )⊗HW = U⊗H(V ⊗HW ). (4.5)

This tells us that ⊗H is commutative and associative, and that the AH-module H acts as an identity element for ⊗H. Since ⊗H is commutative and associative we can define symmetric and antisymmetric products of AH-modules: Definition 4.1.6 [J1, 4.4] Let U be an A -module. Write Nk U for the product H H N0 th U⊗ · · · ⊗ U of k copies of U, with U = . Then the k symmetric group Sk acts H H H H on Nk U by permutation of the U factors in the obvious way. Define Sk U and Λk U H H H to be the A -submodules of Nk U which are symmetric and antisymmetric respectively H H under the action of Sk.

Much of the algebra that works over R or C can be adapted to work over H, using AH-modules and the quaternionic tensor product instead of vector spaces and the real or complex tensor product. There are, however, many situations where the quaternionic tensor product behaves diﬀerently from the standard real or complex tensor product.

For example, the dimension of U⊗HV can behave strangely. It can vary discontinuously 0 0 under smooth variations of U or V , and it is possible to have U⊗HV = {0} when both U and V are non-zero. If φ and ψ are both injective AH-morphisms, it is possible to prove [J1, Lemma 7.4] that φ⊗Hψ is also injective. However, if φ and ψ are both surjective then φ⊗Hψ is not necessarily surjective. Given u ∈ U and v ∈ V it is not possible in general to deﬁne an element u⊗Hv ∈ U⊗HV . However, we do have the following special case:

Lemma 4.1.7 [J1, 4.6] Let U, V be AH-modules, and let u ∈ U and v ∈ V be nonzero. Suppose that α(u)β(v) = β(v)α(u) ∈ H for every α ∈ U † and β ∈ V †. Deﬁne an

45 † ∗ † ∗ element u⊗Hv of H ⊗ (U ) ⊗ (V ) by (u⊗Hv) · (α ⊗ β) = α(u)β(v) ∈ H. Then u⊗Hv is a nonzero element of U⊗HV .

It is easy to visualise how this Lemma ‘works’. If α(u)β(v) = β(v)α(u) ∈ H for every α ∈ U † and β ∈ V †, then α(u) and β(v) must both be in some commutative subﬁeld Cq ⊂ H. This is the same as saying that

† ∗ † ∗ ιU (u) ∈ Cq ⊗ (U ) and ιV (v) ∈ Cq ⊗ (V ) ,

† ∗ and the element u⊗Hv is just the complex tensor product ιU (u) ⊗Cq ιV (v) ∈ Cq ⊗ (U ) ⊗ (V †)∗. So Lemma 4.1.7 tells us that on complex subﬁelds of H, the quaternionic tensor product is the same as the complex tensor product.

4.1.3 Stable and Semistable AH-Modules In this section we define two special sorts of AH-modules, which we shall call semistable and stable. These AH-modules behave particularly well, and we can exploit their ‘nice’ properties to cement further the analogy between real and quaternionic algebra. Definition 4.1.8 [J1, §8] Let U be a finite-dimensional AH-module. We say that U is semistable if it is generated over H by the subspaces U 0 ∩ qU 0 for q ∈ S2.

We can describe semistable AH-modules by the following property:

Lemma 4.1.9 Suppose that U is semistable, with dim U = 4j and dim U 0 = 2j + r, for integers j, r. Then U 0 + qU 0 = U for generic q ∈ S2. Thus r ≥ 0.

The next logical step is to require this property for all q ∈ S2, motivating the following definition: Definition 4.1.10 Let U be a finite-dimensional AH-module. We say that U is stable if U = U 0 + qU 0 for all q ∈ S2.

In effect, our definitions of stable and semistable AH-modules act as a balance to Definition 4.1.1 by demanding that U 0 should not be too small. Many of the properties of semistable and stable AH-modules can be characterised by exploring the properties of a particularly important type of AH-module: 0 Definition 4.1.11 Let q ∈ I \{0}. Define an AH-module Xq by Xq = H, Xq = {p ∈ 0 H : pq = −qp}. In other words, Xq is the subspace of H which is perpendicular to Cq with respect to the standard scalar product.

We quote the following results, mainly taken from [J1, §8]:

• Xq is semistable, but not stable.

• Xq = Xλq for all λ ∈ R \{0}, but for p 6= λq, Xp and Xq are not AH-isomorphic to one another. There is thus a distinct AH-module Xq given by each pair of antipodal points {q, −q} for q ∈ S2.

46 ∼ • There is a canonical AH-isomorphism Xq⊗HXq = Xq, but if p 6= λq then Xp⊗HXq = {0}.

• Let χq : Xq → H be the identity map on H. Then χq and id ⊗ χq : U⊗ Xq → ∼ H H U⊗HH = U are injective AH-morphisms. 0 ∼ 0 0 ∼ n ∼ • There is an isomorphism (U⊗HXq) = U ∩ qU = Cq . It follows that U⊗HXq = nXq.

• Therefore if q ∈ S2 and U is an AH-module with dim U = 4j and dim U 0 = 2j +r, ∼ then U⊗HXq = nXq with n ≥ r . ∼ 2 • If U is semistable then U⊗HXq = rXq for generic q ∈ S , by Lemma 4.1.9. ∼ 2 • An AH-module U is stable if and only if U⊗HXq = rXq for all q ∈ S . ∼ ∼ • It follows that if U and V are stable AH-modules then U⊗HV ⊗HXq = U⊗H(sXq) = 2 rsXq for all q ∈ S , so using the associativity of the quaternionic tensor we infer

that U⊗HV is a stable AH-module.

The AH-module Xq is an important bridge from quaternionic to complex algebra. In ⊥ eﬀect, Xq = Cq ⊕ Cq , and the operation ‘⊗HXq’ converts an AH-module U into copies of Xq, so it turns the AH-module structure on U which is quaternionic information into a set of what are eﬀectively complex vector spaces. It is clear that all stable AH-modules are semistable. There is a sense in which the Xq’s are the ‘only’ class of AH-modules which are semistable but not stable, due to the following Proposition:

Proposition 4.1.12 [J1, 8.8]: Let V be a ﬁnite-dimensional AH-module. Then V is ∼ Ll 2 semistable if and only if V = U ⊕ ( i=1 Xqi ), where U is stable and qi ∈ S .

Joyce also shows that generic AH-modules with appropriate dimensions are stable or semistable:

j Lemma 4.1.13 [J1, 8.9] Let j, r be integers with 0 ≤ r ≤ j. Let U = H and let U 0 be a real vector subspace of U with dim U 0 = 2j + r. For generic subspaces U 0, (U, U 0) is a semistable AH-module. If r > 0 then for generic subspaces U 0, (U, U 0) is a stable AH-module.

The beneﬁts of working with stable and semistable AH-modules become increasingly apparent as one becomes more familiar with the theory. For now, we will quote the following theorems:

Theorem 4.1.14 [J1, 9.1] Let U and V be stable AH-modules with dim U = 4j, dim U 0 = 2j + r, dim V = 4k and dim V 0 = 2k + s. (4.6)

0 Then U⊗HV is a stable AH-module with dim(U⊗HV ) = 4l and dim(U⊗HV ) = 2l + t, where l = js + rk − rs and t = rs .

47 Proof.(Sketch of formula for total dimension) The proof works along the following lines. If dim U = 4j and dim U 0 = 2j + r, then dim(U †)∗ = 2j − r and similarly dim(V †)∗ = 2k − s. So dim(H ⊗ (U †)∗ ⊗ (V †)∗) = 4(2j − r)(2k − s). † ∗ † ∗ Let A = ιU (U) ⊗ (V ) and B = (U ) ⊗ ιV (V ), so that U⊗HV = A ∩ B. Then dim A = 4j(2k − s) and dim B = 4k(2j − r), and dim(A + B) = dim A + dim B − dim(A ∩ B). Now, if r, s ≥ 0 then dim A + dim B ≥ dim(H ⊗ (U †)∗ ⊗ (V †)∗), and so if the subspaces A and B are suitably transverse in H ⊗ (U †)∗ ⊗ (V †)∗ we expect that A + B = H ⊗ (U †)∗ ⊗ (V †)∗, in which case

† ∗ † ∗ dim(U⊗HV ) = dim A + dim B − dim(H ⊗ (U ) ⊗ (V ) ) = 4(js + rk − rs).

The rest of Joyce’s proof consists of showing that if U and V are stable then this 0 intersection is transverse and ﬁnding (U⊗HV ) , from which it is easy to see that U⊗HV is stable. In fact, these dimension formulae still hold if V is only semistable. Theorem 4.1.14 and Proposition 4.1.12 combine to give the following:

Corollary 4.1.15 [J1, 9.3] Let U, V be semistable AH-modules. Then U⊗HV is semistable.

Thus both stable and semistable AH-modules form subcategories of the tensor category of AH-modules, closed under direct and tensor products. Let U be a stable AH-module, with dim U = 4j and dim U 0 = 2j + r. We deﬁne r to be the virtual dimension of U. Then Proposition 4.1.12 shows that the virtual dimension of U⊗HV is the product of the virtual dimensions of U and V . We end this section by quoting the following result:

Proposition 4.1.16 [J1, 9.6] Let U be a stable AH-module, with dim U = 4j and dim U 0 = 2j + r. Let n be a positive integer. Then SnU and Λn U are stable A - H H H modules, with dim(SnU) = 4k, dim(SnU)0 = 2k +s, dim(Λn U) = 4l and dim(Λn U)0 = H H H H 2l + t, where

r+n−1 r+n−1 r+n−1 r−1 r r k = (j−r) + , s = , l = (j−r) + , t = . n − 1 n n n−1 n n

4.2 Duality in Quaternionic Algebra

The objects which are naturally dual to AH-modules are called SH-modules. Under certain circumstances an SH-module can also be regarded as an AH-module. In this case we obtain interesting algebraic results which use dual AH-modules to tell us about AH-morphisms between AH-modules.

48 4.2.1 SH-modules In this section we will describe the class of objects which are dual to AH-modules. These will be called strengthened H-modules, or SH-modules. SH-modules are introduced by Quillen in [Q], as a link between sheaves and AH-modules. Being very much part of quaternionic algebra rather than sheaf theory, we discuss them in this section. Let (U, U 0) be an AH-module. Then u ∈ U is completely determined by the values of α(u) for α ∈ U †, and if u is determined then so is β(u) for all β ∈ U ×. So for all β ∈ U, β(u) is determined by the action of U † on u. Since the only other structure present is the H-action, each β ∈ U × must be an H-linear combination of elements of U †, so U × is generated over H by U †. The converse is clearly true as well — if every β ∈ U × is of the form q · α for some α ∈ U †, then (U, U 0) is an AH-module by Definition 4.1.1. The natural dual to an AH-module is thus an H-module equipped with a generating real subspace. Definition 4.2.1 Let Q be a left H-module and Q† a real linear subspace of Q. We say that the pair (Q, Q†) is a strengthened H-module or SH-module if Q is generated over H by Q†. If U = (U, U 0) is an AH-module then (U ×,U †) is an SH-module, the SH-module corresponding to U. Just as we sometimes write U for the AH-module (U, U 0), we will often write U × for the SH-module (U ×,U †). A further link between these two ideas is provided by choosing a (hyperhermitian) 0 ∼ × metric on the AH-module (U, U ), giving an H-module isomorphism U = U . This in turn identifies U † with (U †)∗, and so realises (U †)∗ as a subspace of U which is perpendicular to U 0. We see that choosing a metric gives us a decomposition U ∼= U 0 ⊕ (U †)∗, where (U, (U †)∗) is an SH-module. Every AH-module can thus be regarded as an SH-module — the point of view depends on whether we think of U 0 or (U 0)⊥ as the ‘special’ subspace. The simplest example is that of the quaternions themselves: we can regard them as the AH-module (H, I) or the SH-module (H, R), and these definitions are exactly equivalent. We define morphisms and quaternionic tensor products for SH-modules, by taking the definitions from their corresponding AH-modules: the whole theory works in exactly the same way. For example, if (U, U 0) and (V,V 0) are AH-modules and φ : U → V is an AH-morphism, then (U ×,U †) and (V ×,V †) are SH-modules and the dual H-morphism φ× : V × → U × satisfies φ×(V †) ⊆ U †, which makes φ× an SH-morphism. We write U × ⊗H V × for the quaternionic tensor product of two SH-modules; in other words we define × H × × U ⊗ V = (U⊗HV ) . (4.7) Another useful example is given by stable and semistable AH-modules. An AH- module is stable (respectively semistable) if and only if it satisfies the identity U = U 0 + qU 0 for all (respectively for generic) q ∈ I. But U 0 + qU 0 = U ⇐⇒ (U 0)⊥ ∩ q(U 0)⊥ = {0}, where (U 0)⊥ is the subspace orthogonal to U 0 with respect to a (hyperhermitian) metric 0 ⊥ ∼ × † on U. Since there is an SH-isomorphism (U, (U ) ) = (U ,U ), we see that U 0 + qU 0 = U ⇐⇒ U † ∩ qU † = {0}.

49 Deﬁnition 4.2.2 An SH-module is called stable (respectively semistable) if and only if it has the property that U † ∩ qU † = {0} for all (respectively for generic) q ∈ I.

This is sometimes easier to demonstrate than the property for the corresponding AH- modules. We shall work happily with either AH-modules or SH-modules according to the needs of each situation, since their theories are interchangeable.

4.2.2 Dual AH-modules In this section we take a new step and ask what happens if we consider (U ×,U †) as an AH-module — the dual AH-module of (U, U 0). There are immediate attractions to this approach. If V is a vector space over the commutative field F then we define the dual space V ∗ to be the space of F-linear maps φ : V → F. In the same way, if U is an H-module we define U × to be the space of H-linear maps φ : U → H. Once we also have a real subspace U 0 ⊂ U we define U † to be the set of maps

† × 0 U = {α ∈ U : α(u) ∈ I for all u ∈ U }.

But an H-linear map α : U → H such that α(U 0) ⊆ I is precisely an AH-morphism from U into H. Thus the space (U ×,U †) consists of two sets of maps φ : U → H, namely the H-linear maps and AH-morphisms respectively. This suggests that deﬁning (U ×,U †) to be the dual AH-module of (U, U 0) could be a good quaternionic analogue of the concept of a dual vector space in real or complex algebra. There is an obvious possible catch: (U ×,U †) might not even be an AH-module! Thus if we are to talk about dual AH-modules, we need to discern which AH-modules have well-deﬁned duals. We have in fact already done this in Section 4.2.1.

Lemma 4.2.3 The AH-module (U, U 0) has a well deﬁned dual AH-module (U ×,U †) if and only if (U, U 0) is also an SH-module.

Proof. In Section 4.2.1 we showed that (U, U 0) is an AH-module if and only if (U ×,U †) is an SH-module. We simply reverse this argument: (U ×,U †) is an AH-module if and only if the AH-module (U, U 0) is also an SH-module.

Deﬁnition 4.2.4 Let U be an H-module and U 0 a real subspace of U . We say that the pair (U, U 0) is a strengthened augmented H-module, or SAH-module, if (U, U 0) is both an AH-module and an SH-module.

A comprehensive way to sum this up is to say that (U, U 0) is an AH-module if and only if it has no submodule isomorphic to (H, H) , an SH-module if and only if it has no submodule isomorphic to (H, {0}), and an SAH-module if and only if it has no submodule isomorphic to either (H, H) or (H, {0}). Unless otherwise stated, when we refer to properties of an SAH-module such as stability, we mean this in terms of the structure of U as an AH-module. Example 4.2.5 Any semistable AH-module U has the property that U = U 0 + qU 0 for generic q ∈ I, so U is also an SH-module and so an SAH-module.

50 Deﬁnition 4.2.6 Let U = (U, U 0) be an AH-module which is also an SAH-module. Then we deﬁne U × = (U ×,U †) to be the dual AH-module of U.

With this definition, it is clear that U × is also an SAH-module. For finite dimensional U, there are canonical isomorphisms U ∼= (U ×)× and (U ×)† ∼= U 0. Definition 4.2.7 Let (U, U 0) be an AH-module. Then U is called antistable if and only if U 0 ∩ qU 0 = {0} for all q ∈ I.

It is clear that antistable AH-modules are almost always dual to stable AH-modules. The only irreducible exception is the AH-module (H, {0}), which we regard as an antistable AH-module in spite of the fact that its dual (H, H) is not an AH-module.

4.2.3 Spaces of AH-morphisms and Duality The space U † is, as we have remarked, the space of AH-morphisms φ : U → H. This fact is an example of a more general result which makes the theory of dual spaces in quaternionic algebra particularly useful. Let A and B be (free, ﬁnite dimensional) modules over the commutative ring R, and let HomR(A, B) denote the space of R-linear ∼ maps φ : A → B. It is well-known that there is a canonical isomorphism HomR(A, B) = ∗ A ⊗R B. There is an analogous result in quaternionic algebra. We start with the following deﬁnition:

Deﬁnition 4.2.8 Let U and V be AH-modules. Then HomAH(U, V ) denotes the space of AH-morphisms from U into V . Here is the main result of this section:

Theorem 4.2.9 Let U be an SAH-module and V be an AH-module. Then there is a canonical isomorphism ∼ × 0 HomAH(U, V ) = (U ⊗HV ) .

× 0 Proof. Let φ ∈ (U ⊗HV ) . Then × † ∗ 0 ∗ 0 ∗ † ∗ φ ∈ (ιU × (U ) ⊗ (V ) ) ∩ ((U ) ⊗ ιV (V )) ∩ (I ⊗ (U ) ⊗ (V ) ), or equivalently † † ∗ 0 ∗ 0 φ ∈ (ιU × (U ) ⊗ (V ) ) ∩ ((U ) ⊗ ιV (V )). † † ∗ † † Consider φ ∈ ιU × (U ) ⊗ (V ) . The mapping ιU × identiﬁes U with ιU × (U ). Using this and the canonical isomorphism of real vector spaces Hom(A, B) ∼= A∗ ⊗ B we see that φ is exactly equivalent to an real linear map

Φ:(U †)∗ → (V †)∗, which in turn is equivalent to an H-linear map

† ∗ † ∗ ΦH : H ⊗ (U ) → H ⊗ (V ) ,

51 which by deﬁnition is an AH-morphism. Every AH-morphism ψ : U → V is equivalent to an AH-morphism Ψ : H ⊗ (U †)∗ → † ∗ H ⊗ (V ) with the property that Ψ : ιU (U) → ιV (V ). Using the fact that φ ∈ 0 ∗ 0 0 ∼ 0 (U ) ⊗ ιV (V ) and the natural identiﬁcation U = ιU (U ) we see that

0 0 ΦH : ιU (U ) → ιV (V ).

0 Since U is an SAH-module, ιU (U) is generated over H by ιU (U ), from which it follows that ΦH : ιU (U) → ιV (V ). Thus ΦH is equivalent to an AH-morphism from U to × 0 V . Reversing these steps, we can construct an element of (U ⊗HV ) from each AH- morphism from U to V . The quaternionic tensor product can be used in this way to tell us about spaces of AH-morphisms, which is another piece of evidence suggesting that Joyce’s deﬁnition of the quaternionic tensor product is the right one for AH-modules.

4.3 Real Subspaces of Complex Vector Spaces

0 ∼ n We have seen that choosing different real subspaces U of an H-module U = H gives rise to different algebraic properties. As always with the quaternions, it is useful to compare this situation with that of the complex numbers. As an instructive example (and a bit of light relief!) we shall give a classification of real linear subspaces of complex vector spaces up to complex linear isomorphism. In other words, we classify the possible k n orbits of a subspace R of C under the action of GL(n, C). This is not difficult, though as far as the author can tell both the problem and its solution are original.

0 ∼ k n Theorem 4.3.1 Let U = R be a real linear subspace of C . Then we can choose a n complex basis {ej : j = 1, . . . , n} of C such that

0 1 p 1 q U = he , . . . , e , ie , . . . , ie iR, where q ≤ p ≤ n, p + q = k.

0 0 0 0 n 0 0 ∼ p Proof. Both U + iU and U ∩ iU are complex subspaces of C . Let U + iU = C 0 0 ∼ q 0 and let U ∩ iU = C . Then dimR(U ) = p + q. Choose a complex basis {e1, . . . , eq} for U 0 ∩ iU 0. Then {e1, ie1, . . . , eq, ieq} is a real basis for U 0 ∩ iU 0. Extend this to a real basis {e1, ie1, . . . , eq, ieq, eq+1, . . . , ep} for 0 1 p 0 0 ∼ p 1 p U . Then {e , . . . , e } spans U + iU = C (over C), and so {e , . . . , e } is linearly independent over C. The result follows. n It is easy to see from this theorem that choosing a real subspace of C is always compatible with the complex structure, in the sense that each basis vector of the real 0 0 ∼ subspace U can be chosen to lie in one and only one copy of C. The pair (U, U ) = n k (C , R ) can always be completely reduced to a direct sum of copies of C, each of which contains 0, 1 or 2 basis vectors for U 0. The situation is very diﬀerent for H-modules. For example, consider the AH-module (U, U 0) with 2 0 U = H and U = h(1, 0)(0, 1), (i1, i2)i.

52 There is no way to decompose U into two separate copies of H, the first of which contains 2 basis vectors for U 0 and the second of which contains the remaining basis vector. This motivates the following definition: Definition 4.3.2 An AH-module (U, U 0) is irreducible if and only if it cannot be written as a direct sum of two non-trivial AH-modules, i.e. there are no two non-trivial 0 0 0 0 0 AH-modules (U1,U1), (U2,U2) such that (U, U ) = (U1 ⊕ U2,U1 ⊕ U2).

The classiﬁcation of irreducible AH-modules up to AH-isomorphism is a much more diﬃcult problem than its analogue for complex vector spaces. It will be addressed in the next section using a class of algebraic objects called K-modules.

4.4 K-modules

A K-module is an algebraic object based on a pair of complex vector spaces. Real and quaternionic vector spaces, as so often, occur as complex vector spaces with particular structure maps. Definition 4.4.1 [Q, 4.1] A K-module is a pair (W, V ) of (finite dimensional) complex ∼ 2 vector spaces together with a linear map e : W → H ⊗ V , where H = C is the basic representation of GL(2, C). The reason why K-modules are important to quaternionic algebra is that in the presence of suitable structure maps, some K-modules are equivalent to AH-modules. This allows us to use the classification of irreducible K-modules, a problem with a known solution, to write down all irreducible AH-modules very explicitly. This link between K-modules and AH-modules was discovered by Quillen [Q]. Quillen is more interested in an interpretation of quaternionic algebra in terms of sheaves over the Riemann sphere, a powerful theory which we will review in the next section. We follow a slightly different approach from that in Quillen’s paper to obtain a more immediate link between K- modules and AH-modules. A K-module e : W → H ⊗ V is called indecomposable if it cannot be written as the e1 direct sum of two non-trivial K-modules. A K-module morphism is a map φ :(W1 → e2 H ⊗ V1) −→ (W2 → H ⊗ V2) which respects the K-module structure. A K-module can equivalently be defined as a pair of linear maps e1, e2 : W → V . In this guise, a K-module is a representation of the Kronecker quiver. We recover the first definition by setting e(w) = h1 ⊗e1(w)+h2 ⊗e2(w) where {h1, h2} is a basis for H. Representations of the Kronecker quiver are discussed in Benson’s book [Ben, §4.3]. The important result is the following classification theorem of Kronecker (which we have summarised slightly), which shows that every indecomposable K-module is isomorphic to one of three basic types.

Theorem 4.4.2 [Ben, p. 101] Let e1, e2 : W → V be a pair of linear maps constituting an indecomposable K-module. Then one of the following holds:

(i) The vector spaces W and V have the same dimension. In this case, if det e1 6= 0

53 then e1 and e2 can be written in the form

 α 0   1 α    e1 7→ id e2 7→  .. ..  .  . .  0 1 α

If det e1 = 0 a modiﬁcation is necessary which in some sense corresponds to the ‘rational canonical form at inﬁnity’. (ii) The dimension of W is one larger than the dimension of V , and bases may be chosen so that e1 and e2 are represented by the matrices

 1 0 0   0 1 0   .. .   . ..  e1 =  . .  e2 =  . .  . 0 1 0 0 0 1

(iii) The dimension of W is one smaller than the dimension of V , and bases may be chosen so that e1 and e2 are represented by the transposes of the above matrices.

n,α Definition 4.4.3 Define X1 to be the indecomposable K-module of type (i) with n,∞ dim V = n and α on the leading diagonal of e2. (We write X1 for the case det e1 = n n 0 ). Define X2 to be the irreducible K-module of type (ii) with dim V = n. Define X3 to be the irreducible K-module of type (iii) with dim V = n.

Let σH be the standard quaternionic structure map on H defined by σH (z1h1 + z2h2) = −z¯2h1 +z ¯1h2. This gives H ⊗ V the structure of a complex H-module. Our aim is for the K-module e : W → H ⊗ V to define a real subspace of a real H-module. This is accomplished by compatible structure maps on W and V . Definition 4.4.4 [Q, 11.2] An SK-module is a K-module e : W → H ⊗V equipped with antilinear operators σW and σV of squares 1 and −1 respectively, such that e · σW = (σH ⊗ σV ) · e.

Suppose W → H ⊗ V is an SK-module. Then σW is a real structure on W , and its σ set of ﬁxed points is the real vector space W . The map σH ⊗σV is also a real structure, and in the same way we deﬁne the real vector space (H ⊗ V )σ. Then (H ⊗ V )σ is a real H-module, whose H-module structure is inherited from that on H, and e(W σ) is a real subspace of this H-module. In many circumstances an SK-module is therefore equivalent to an AH-module. 2 Example 4.4.5 Consider the K-module X2 which has dim W = 3 and dim V = 2. Let {w1, w2, w3} be a basis for W and let {v1, v2} be a basis for V . We have

e(w1) = h1 ⊗ v1, e(w2) = h1 ⊗ v2 + h2 ⊗ v1, e(w3) = h2 ⊗ v2.

Let σV be the standard quaternionic structure on V , so σV (v1) = v2 and σV (v2) = −v1. There is a compatible real structure σW on W given by σW (w1) = w3, σ(w3)W = w1

54 2 and σW (w2) = −w2, so that with these structure maps X2 is an SK-module. We have real vector spaces σ W = hw1 + w3, iw2, i(w1 − w3)i and h ⊗ v − h ⊗ v i(h ⊗ v + h ⊗ v ) (H ⊗ V )σ = 1 2 2 1 1 2 2 1 . h1 ⊗ v1 + h2 ⊗ v2 i(h1 ⊗ v1 − h2 ⊗ v2) σ ∼ An H-module isomorphism (H ⊗V ) = H is obtained by mapping these basis vectors to σ 1, i1, i2 and i3 respectively. Under this isomorphism the real subspace e(W ) is mapped 2 to the imaginary quaternions I. This demonstrates explicitly that the SK-module X2 is equivalent to the AH-module (H, I ) = H. Every real subspace U 0 of a quaternionic vector space U can be obtained in this fashion, so a classiﬁcation of SK-modules gives a classiﬁcation of AH-modules. Corollary 4.4.6 Every indecomposable SK-module is isomorphic to one of the following: n,α n,−α¯−1 (A) The direct sum of a pair of K-modules of type (i) of the form X1 ⊕ X1 . (B) An indecomposable K-module of type (ii) or (iii) with dim V even; in other words 2m 2m X2 or X3 . (C) An indecomposable K-module of type (ii) or (iii) with dim V odd, tensored with the basic representation H equipped with its standard structure map σH ; in other words 2m+1 2m+1 X2 ⊗ H or X3 ⊗ H.

Proof. This follows from Theorem 4.4.2 and explicit calculations using standard structure maps on the vector spaces V . It remains to check which pairs (U, U 0) arising in this fashion are AH-modules. As noted earlier, a pair (U, U 0) fails to be an AH-module if and only if it has a subspace 1 of the form (H, H). This pair is given by the SK-module H ⊗ X3 . The ‘annihilating 0 K-module’ X3 also fails to give an AH-module. Any SK-module containing neither of these indecomposables is equivalent to an AH-module. Here are some important facts about irreducible AH-modules which can now be deduced:

• SK-modules of type (ii) correspond to stable AH-modules and SK-modules of type (iii) correspond to antistable AH-modules.

1,α 1,−α¯−1 • Indecomposable SK-modules of type (i) of the form X1 ⊕ X1 correspond to the semistable AH-modules Xq. 2m • The irreducible stable AH-module corresponding to X2 has dim U = 4m and 0 ∼ m dim U = 2m + 1. Thus U = H and the virtual dimension of U is 1. 2m+1 • The irreducible stable AH-module of the form H ⊗ X2 has dim U = 4(2m + 1) 0 ∼ 2m+1 and dim U = 4(m + 1). Thus U = H and the virtual dimension of U is 2. • This shows that there is an irreducible stable AH-module with virtual dimension 1 in every dimension and an irreducible stable AH-module with virtual dimension 2 in every odd dimension.

55 • The isomorphism class of an irreducible stable AH-module is thus uniquely determined by the dimension and virtual dimension of U.

4.5 The Sheaf-Theoretic approach of Quillen

Much of Joyce’s quaternionic algebra can be described using (coherent) sheaves over 1 the complex projective line CP . This interpretation is due to Daniel Quillen [Q]. Quillen’s paper works by recognising that certain exact sequences of sheaf cohomology groups are K-modules. Thus in the presence of certain structure maps, we obtain SH- modules. (Quillen deals primarily with SH-modules rather than AH-modules.) Quillen uses slightly diﬀerent structure maps from those we used in the previous section to obtain SK-modules, but the resulting theory is exactly the same. The most interesting new result in this section is that the equivalence between sheaves and SH-modules respects tensor products, enabling us to calculate the quaternionic tensor product of two SH-modules from knowing the tensor products of the corresponding sheaves. Thus by the end of this section we will have succeeded in classifying all AH- modules and their tensor products.

4.5.1 Sheaves on the Riemann Sphere 1 We describe the algebraic geometry of (coherent) sheaves over CP . 1 Quillen demon- 1 strates that every coherent sheaf over CP is the direct sum of a holomorphic vector bundle and a torsion sheaf (one whose support is ﬁnite). 2 These summands factorise very neatly — every torsion sheaf is the sum of indecomposable sheaves supported at a single point, and every holomorphic vector bundle is a sum of holomorphic line bundles. We will describe the vector bundles ﬁrst. n Every holomorphic line bundle over CP is a tensor power of the hyperplane section 1 bundle L [GH, p. 145]. In the case n = 1, we use the open cover of CP = C ∪ {∞} 1 1 consisting of the two open sets U0 = CP \{∞} and U1 = CP \{0}. A holomorphic line 1 ∗ bundle over CP is determined by a holomorphic transition function ψ : U0 ∩ U1 → C , ∗ ∗ so ψ : C → C . Two transition functions ψ and ψ0 determine the same line bundle if ∗ ∗ and only if there exist non-vanishing holomorphic functions f, g : C → C such that f ψ0 = ψ. g Two functions are equivalent under this relation if and only if they have the same winding 1 number, so each line bundle on CP is given by one of the transition functions g(z) = zn, n Nn n ∈ Z. The line bundle given by the transition function z is in fact L. Following standard notation, we write O(n) for the sheaf of its holomorphic sections. Thus O =

1Background material can be found in [GH], which introduces sheaves and their cohomology [pp. 34-49], coherent sheaves [pp. 678-704], holomorphic vector bundles [pp. 66-71] and holomorphic line bundles [pp. 132-139]. A more thorough exposition of the diﬀerential geometry of holomorphic vector bundles, including many of the properties of sheaves used in Quillen’s paper, can be found in Kobayashi’s book [K]. 2This uses the convention of identifying a holomorphic vector bundle with its sheaf of holomorphic sections.

56 1 O(0) is the structure sheaf of CP . It is easy to see (by multiplying the transition n m ∼ n+m ∼ functions together) that L ⊗ L = L , or in sheaf-theoretic terms O(n) ⊗O O(m) = O(n + m). 1 Every holomorphic vector bundle over CP can be written as a direct sum of these line bundles, the summands being unique up to order. 3 Thus every holomorphic vector L ∞ n bundle E is a sum of irreducible line bundles, and can be written E = −∞ anL , where the multiplicities an are unique (though the decomposition itself may not be). This leaves us to consider sheaves which are supported at a finite set of points, which are called torsion sheaves. Quillen [Q, §2] demonstrates that every coherent sheaf over 1 CP is the sum of a vector bundle and a torsion sheaf. Torsion sheaves themselves 1 split into sheaves supported at one point only. Let z ∈ CP be such a point, and let Oz be the ring of germs of holomorphic functions at z. Define mz to be the unique maximal ideal of Oz consisting of germs of functions whose first derivative vanishes at n n z. Every torsion sheaf splits into sheaves of the form Oz/(mz) , which we write O/mz by extending mz by O on the complement of z. We have the following Theorem: 1 Theorem 4.5.1 [Q, 2.3] Any coherent sheaf over CP splits with unique multiplicities n into indecomposable sheaves of the form O(n) for n ∈ Z and O/mz for n ≥ 1 and 1 z ∈ CP .

Cohomology Groups and Exact Sequences There are various ways to calculate the cohomology groups of these sheaves. 4 The method Quillen outlines uses the properties of exact sequences of sheaves. It is from the maps in these sequences that we obtain K-modules and thence SH-modules. ∼ 2 1 Let H = C be the basic representation of GL(2, C). Then CP can be identiﬁed with the set of quotient lines of H and there is a basic exact sequence

0 → Λ2H ⊗ O(−1) → H ⊗ O → O(1) → 0. (4.8)

2 ∼ Tensoring (over O ) with the sheaf F and choosing an identiﬁcation Λ H = C yields the exact sequence 0 → F (−1) → H ⊗ F → F (1) → 0, (4.9) where F (n) denotes the sheaf F ⊗O O(n). Example 4.5.2 If we put F = O(n), n ≥ 0, we have the exact sequence

0 → O(n − 1) → H ⊗ O(n) → O(n + 1) → 0. (4.10)

0 ∼ We know that H (O) = C (global holomorphic functions on CP (1) ) and that H0(O(−1)) = 0. By the exact sequences of (4.10) and induction, it follows that H0(O(n)) ∼= Sn(H) for n ≥ 0 and zero otherwise, and there is an exact sequence of cohomology groups given by

0 → Sn−1H → H ⊗ SnH → Sn+1H → 0. (4.11)

3This follows from the Harder-Narasimhan ﬁltration of a holomorphic vector bundle E over any Riemann surface M [K, p. 137] 4For example [W, p.11], where H0(O(n)) is shown to be isomorphic to the (complex) vector space of homogeneous polynomials of degree n in 2 variables.

57 Torsion sheaves can be dealt with in a similar fashion, using a resolution involving 0 0 n ∼ n the sheaf cohomology groups H (F ). It is easy to see that H (O/mz ) = C . We call the sheaves O(n) where n ≥ 0, torsion sheaves, and sums thereof regular sheaves. For all regular sheaves F , the first cohomology group H1(F ) is zero. This leaves the sheaves O(n) where n < 0 and sums thereof, which we call negative vector bundles. These must be treated slightly differently, using the first cohomology groups H1(F ). Since H0(O(n)) = 0 for n < 0, we obtain an exact sequence

0 → H1(O(n − 1)) → H ⊗ H1(O(n)) → H1(O(n + 1)) → 0, (4.12) from which it follows that H1(O(n)) ∼= S−n−2H for n ≤ −2 and zero otherwise.

Sheaves and K-modules Consider the regular sheaf O(n) for n ≥ 0. The exact sequence (4.10) gives rise to an injection of cohomology groups H0(O(n − 1)) → H0(H ⊗ O(n)) which takes the form e : Sn−1H → H ⊗ SnH. This is clearly a K-module, the irreducible K-module n+1 n 0 n X3 . Similarly for the torsion sheaves O/mx there is an injection H (O/mx(−1)) → 0 n H (H ⊗ O/mx). This gives a K-module of type (i) and dim W = n. Thus we obtain a K-module from each regular sheaf F which we call ξ+F . Let O(n), n < 0 be a negative vector bundle. The exact sequence (4.12) gives a map H1(O(n − 1)) → H ⊗ H1(O(n)) which is equivalent to the indecomposable K-module −n−1 X3 . Thus for any negative vector bundle G we obtain a K-module which we call ξ−G. Comparing the classifications of indecomposable sheaves and K-modules (Theorems 4.4.2 and 4.5.1), it is clear that these categories are equivalent. Quillen proves this in detail [Q, §§4,5] and uses the tensor product of sheaves F ⊗O G to construct an equivalent tensor product operation for K-modules [Q, §6]. The rest of the programme begins to take shape. Some sheaves will correspond to SK- modules, from which we obtain SH-modules. Quillen formulates this slightly differently from our treatment in Section 4.4. Let W → H ⊗ V be a K-module. Instead of a real structure on W and a quaternionic structure on V , Quillen uses K-modules with a quaternionic structure σW on W and a real structure σV on V , such that the map e intertwines σW and σH ⊗ σV . In this situation, W and H ⊗ V are H-modules and e is an H-linear map. He calls this structure a σK-module.A σK-module is not itself an SH-module, but the inclusion of the real subspace V σ in the cokernel (H ⊗ V )/W is an SH-module if the K-module has no submodule for which the map e is surjective, in which case the K-module is called reduced. There is a parallel description in terms of sheaves. Let σ : z 7→ z¯−1 be the antipodal map on the Riemann sphere. This induces a map of sheaves σ∗ : F 7→ σ∗(F ) which we call the σ-transform, and allows us to define a ‘ σ-invariant sheaf’ or just ‘ σ-sheaf’. For example, a torsion sheaf F is a σ-sheaf if it is supported at a finite set of points which is preserved by the antipodal map σ — so it must consist of sheaves of the n n ∗ form O/mz ⊕ O/mσ(z), where σ interchanges the two summands. Quillen investigates σ-sheaves thoroughly, and discovers that:

58 Proposition 4.5.3 [Q, 10.7] Any σ-sheaf splits with unique multiplicities into the following irreducible σ-sheaves: n (1) O/(mzmσ(z)) for any pair {z, σ(z)} of antipodal points and n ≥ 1. (2) O(2m) for m ∈ Z. (3) O(2m + 1) ⊗ H for m ∈ Z.

The formal similarity between this result and Corollary 4.4.6 is clear. He also proves that:

Proposition 4.5.4 [Q, 12.6] The categories of reduced σK-modules and SH-modules are equivalent.

We can also use σ-sheaves to obtain SK-modules which lead directly to SH-modules. Example 4.5.5 Let F be a regular σ-sheaf. Then we have the exact sequence

0 → H0(F ) → H ⊗ H0(F (1)) → H0(F (2)) → 0, (4.13) and H0(F ) → H ⊗ H0(F (1)) is an SK-module e : W → H ⊗ V . Taking real subspaces gives an SH-module which we call η+(F ). Example 4.5.6 Let G be a negative σ-vector bundle with no summand H ⊗ O(−1) or O(−2). Then we have the exact sequence

0 → H1(G) → H ⊗ H1(G(1)) → H1(G(2)) → 0, (4.14)

Just as in the previous example, this sequence gives an SK-module

H1(G) → H ⊗ H1(G(1)). (4.15)

We call this SH-module η−(G). We also deﬁne η−(O(−2)) = η−(H ⊗ O(−1)) = 0. If we have a σ-sheaf A = F +G, with F and G as above, then H0(G) = H0(G(1)) = 0 and H1(F ) = H1(F (1)) = 0 ; so η+(G) = η−(F ) = 0. In theory, we could combine the functors η+ and η− into a single functor η = η+ + η−, since η(A) = η+(F ) + η−(G) as required.

4.5.2 Sheaves and the Quaternionic Tensor Product We have seen how the tensor product of sheaves encourages us to deﬁne a ‘reduced tensor product’ operation for K-modules. It turns out that this tensor product agrees remarkably with the quaternionic tensor product for SH-modules. This is a considerable bonus from Quillen’s theory — the correspondences between σ-sheaves, SK-modules and SH-modules allow us to compute tensor products in each category. The main theorem is as follows:

Theorem 4.5.7 [Q, 7.1] 5

5Quillen proves this theorem for the tensor product of K-modules — the version given here is obtained by performing the simple translation into SH-modules.

59 Let Fi be regular σ-sheaves and Gi be negative σ-vector bundles. Then

+ H + + η F1 ⊗ η F2 = η (F1 ⊗O F2),

+ H − − η F1 ⊗ η G1 = η (F1 ⊗O G1) and − H − η G1 ⊗ η G2 = {0}.

Since we will usually work with AH-modules, we will often ﬁnd ourselves using this theorem for the corresponding AH-modules, which of course take the form (η+F )× and (η−G)×. + × Thus if F is a torsion σ-sheaf and G is a negative σ vector bundle, (η F ) ⊗H (η−G)× = {0}. If F = O(m) and G = O(n) (possibly tensored with H if m or n is odd) with m ≥ 0 and n < −2 then + × − × 0 m + n ≥ −2 (η F ) ⊗H(η G) = − × η (F ⊗O G) m + n ≤ −3 since η−(O(k)) = {0} for k ≥ −2. The sheaf-theoretic approach is thus a very powerful tool for describing quaternionic algebra. For example, it is possible to obtain the dimension theorems of Section 4.1.3 by translating known results about the degree and rank of the tensor product of two sheaves.

60 Chapter 5

Quaternionic Algebra and Sp(1)-representations

The representations of the group Sp(1) occur in so many different situations, from Kähler geometry to particle physics, that they are by far the most ubiquitous Lie group representations in modern mathematical literature. Given this versatility, it is no surprise that these representations are a powerful tool in quaternionic algebra, especially since Sp(1) is just the group of unit quaternions and the Lie algebra sp(1) can be identified with I. In this chapter, we will see how stable and antistable AH-modules can be handled using Sp(1)-representations. We encountered the germ of this idea in Section 3.4, where we came across the space H ⊗ Ek,r and its splitting into two H-submodules. Because 1 the Riemann sphere CP can be described as the homogeneous space Sp(1)/U(1), the 1 holomorphic sections of line bundles over CP are naturally Sp(1)-representations. The structure maps necessary to define an SK-module e : W → H ⊗ V of type (ii) or (iii) arise from Sp(1)-representations on W , H and V . Not only do Sp(1)-representations underlie all of these phenomena — they also make the theory of quaternionic algebra very easy to predict and manipulate. This point of view turns out to have fruitful applications in hypercomplex geometry, towards which our exposition is deliberately geared. We use representations to explain the structure of stable AH-modules and their tensor products. We also investigate the role of semistable AH-modules and their interaction with the Sp(1)-representation structure of stable AH-modules.

5.1 Stable AH-modules and Sp(1)-representations 5.1.1 Sp(1)-representations on the quaternions As a motivating example, we review the case of the quaternions themselves, viewed as the stable AH-module (H, I). Recall the description of the quaternions as a tensor product ∼ of two Sp(1)-representations H = V1 ⊗ V1, given in Equation (3.17), where the left hand

61 1 copy of V1 gives the left H-action, and the right-hand copy gives the right H-action. In other words, we think of H as an Sp(1) × Sp(1)-representation by deﬁning

−1 (p, q): r 7→ prq r ∈ H, p, q ∈ Sp(1) ⊂ H. Consider now the action of the diagonal Sp(1)-subgroup {(q, q): q ∈ Sp(1)} ⊂ ∼ Sp(1) × Sp(1) on V1 ⊗ V1. The Clebsch-Gordon formula gives the splitting V1 ⊗ V1 = ∼ 2 2 V2 ⊕ V0 (equivalent to the standard isomorphism V ⊗ V = S V ⊕ Λ V ). Each of these summands inherits a real structure from the real structure on V1 ⊗ V1 so we obtain the splitting ∼ V1 ⊗ V1 = V2 ⊕ V0 (5.1) into real subspaces of dimensions three and one respectively, just as we would expect. This is the same as taking the action by conjugation r 7→ qrq−1, which as we know ∼ 0 ∼ preserves the splitting H = I⊕R. For the quaternions, the AH-module structure H = I † ∗ ∼ and (H ) = R is a concept which arises naturally when we take both the Sp(1) actions into account. It is this account of the AH-module H which we will generalise to all stable AH-modules. This description of the quaternions is very similar to that of Example 4.4.5, where ∼ 2 H = X2 . In terms of Sp(1)-actions, the basic vector space H we used so much in the previous chapter is simply a copy of the basic representation V1.

5.1.2 Notation for Several Sp(1)-representations It will be a sound investment at this point to introduce some notation to help us keep track of the structure of representations when we have several copies of Sp(1) acting on a vector space. We have already encountered the action of Sp(1) × Sp(1) on V1 ⊗ V1. Here we have two copies of Sp(1) acting, so there is already the possibility of ambiguity concerning which Sp(1) is acting on which V1. We remove this ambiguity by writing upper-case superscripts with the groups and the representations, to make it clear which group is acting on which vector space. For left H-modules there will always be a left H-action to consider. We will denote L L this by V1 , and the copy of Sp(1) which acts on this factor by Sp(1) . Other copies of Sp(1) and other representations will be labelled with the letters M,N etc. So we would write the above example as

L M L M Sp(1) × Sp(1) acting on V1 ⊗ V1 .

When we decompose such a representation using the Clebsch-Gordon formula, we are decomposing the action of the diagonal subgroup {(q, q)} ⊂ Sp(1)L × Sp(1)M . We will call this subgroup Sp(1) LM , thus stating explicitly of which two groups this is the diagonal subgroup. Similarly, we can combine superscripts for the representations to write L M ∼ LM LM V1 ⊗ V1 = V2 ⊕ V0 .

1 4 We are talking about the representation V1 ⊗ V1 as a real representation on R , implicitly using the induced map σ1 ⊗ σ1 as a real structure on V1 ⊗ V1. For more details, refer to Section 1.2.1.

62 This book-keeping comes into its own when we come to consider tensor products of many Sp(1)-representations. For example, if we have three copies of Sp(1) acting, we write this as

L M N L M N Sp(1) × Sp(1) × Sp(1) acting on V1 ⊗ Vj ⊗ Vk . In this situation there are various diagonal actions we could be interested in, and we can join the superscripts as above to indicate exactly which one we are considering. For example, supposing we want to restrict to the diagonal subgroup in the ﬁrst two copies of Sp(1), i.e. {(q, q)} × Sp(1)N . We denote this subgroup Sp(1)LM × Sp(1)N . We combine superscripts for the representations in the same way, so that we now have

LM N LM LM N Sp(1) × Sp(1) acting on (Vj+1 ⊕ Vj−1 ) ⊗ Vk . If, however, we considered the diagonal subgroup of the ﬁrst and last copies of Sp(1), we would write this as

LN M LN LN M Sp(1) × Sp(1) acting on (Vk+1 ⊕ Vk−1) ⊗ Vj . This provides an unambiguous and (it is hoped) easy way to understand tensor products of several representations and their decompositions into irreducibles under diﬀerent diagonal actions.

5.1.3 Irreducible stable AH-modules We have described the quaternions as an Sp(1)L × Sp(1)M -representation. This allows 0 ∼ us to interpret the primed part H = I as a representation of the diagonal subgroup of Sp(1)LM ⊂ Sp(1)L × Sp(1)M . In this section we will demonstrate how this idea can be adapted to describe more general stable AH-modules. The basic idea is exactly the L M L M same — a stable AH-module is a (real) Sp(1) × Sp(1) -representation V1 ⊗ W , M Lk M where W = 1 ajVj . The left H-action is given by the action of the left subgroup Sp(1)L ⊂ Sp(1)L × Sp(1)M . The primed part is then a representation of the diagonal subgroup Sp(1)LM ⊂ Sp(1)L × Sp(1)M . Let (U, U 0) be an irreducible stable (or antistable) AH-module. Let M be the group of AH-automorphisms of U whose (real) determinant is equal to 1. Using Theorem 4.2.9, we see that M = ±1 if the virtual dimension U is 1, and M ∼= Sp(1) if the virtual dimension of U is 2. Consider now the more general group G of real linear isomorphisms φ : U → U such that:

• φ(U 0) = U 0, • The (real) determinant of φ is 1,

• There exists some q ∈ Sp(1) such that φ(pu) = (qpq−1)φ(u) for all p ∈ H, u ∈ U. Then G is a compact Lie group of which M is a normal subgroup. Because of this the Lie algebra of G splits into two orthogonal ideals

g = sp(1) ⊕ m

63 and the exponential map determines a homomorphism ρ : Sp(1) → G. Since the elements of G map U 0 to itself, ρ is a representation of Sp(1) on U 0. Since every stable AH-module is a sum of such irreducibles, there is an action of Sp(1) on U 0 for all stable AH-modules U, and the irreducible decomposition of U as an AH-module determines the irreducible decomposition of the Sp(1)-action on U 0, and vice versa. As we shall see, there is a unique irreducible stable AH-module for each irreducible Sp(1)-representation. Let U be a stable AH-module and suppose that U 0 is preserved by some diagonal action of Sp(1). This diagonal action will be the result of the left H-action on V1 and some other Sp(1)-action on U. Thus our AH-modules will follow the basic form L M U = V1 ⊗ Vm (5.2) as a representation of the group Sp(1)L × Sp(1)M . The primed part of U is then the LM Vn+1 summand in the decomposition L M ∼ LM LM V1 ⊗ Vn = Vn+1 ⊕ Vn−1 . For example, recall the splitting ∼ n H ⊗ Ek,r = V1 ⊗ εk,r(Vr+1 ⊕ Vr−1) from Section 3.4. We see that each of these summands is an AH-submodule of H ⊗ Ek,r. The larger (left-hand) submodule is stable; the smaller one is antistable. Both stable and antistable AH-modules arise as Sp(1) × Sp(1)-representations. This is not necessarily the case for AH-modules which are neither stable nor antistable. For example, there is no representation of Sp(1) on Xq which couples with the 0 left H-action to give a representation of Sp(1) on Xq.

AH-modules of the form V1 ⊗ V2m−1

Consider an even-dimensional irreducible Sp(1)-representation V2m−1. Because σ = L M ∼ 4m σ1 ⊗ σ2m−1 is a real structure, there is a real representation V1 ⊗ V2m−1 = R with a left H-action deﬁned by q : a ⊗ b 7→ (qa) ⊗ b. Under the diagonal Sp(1)LM -action q : a ⊗ b 7→ (qa) ⊗ (qb), we have the splitting L M ∼ LM LM V1 ⊗ V2m−1 = V2m ⊕ V2m−2, (5.3) L M σ ∼ which is a splitting of real vector spaces (technically we could write (V1 ⊗ V2m−1) = LM σ LM σ (V2m ) ⊕ (V2m−2) ).

L M LM 0 Proposition 5.1.1 The pair (V1 ⊗ V2m−1,V2m ) forms a stable AH-module (U, U ) ∼ m 0 ∼ 2m+1 with U = H and U = R .

Proof. Consider the maximal stable submodule W of U. Since W is an H-submodule it must be invariant under the left H-action. Also, W must depend solely on the Sp(1) × Sp(1)-representation structure: in particular W 0 must be preserved by the 0 LM 0 diagonal action. So W must be an Sp(1) -invariant subspace of U = V2m and 0 0 0 1 by Schur’s Lemma [FH, p.7] W = V2m or W = {0}. Since dim U > 2 dim U we must have W 6= {0}. Hence (W, W 0) = (U, U 0) and thus U is stable.

64 Lemma 5.1.2 The AH-module U = V1 ⊗ V2m−1 is the irreducible stable AH-module 2m corresponding to the SK-module X2 .

Proof. This follows from Proposition 5.1.1 and the remarks in Section 4.4. The virtual dimension of U is 1, from which it follows that U must be irreducible. The isomorphism 2m with X2 follows because irreducible stable AH-modules are uniquely determined by their dimensions.

It is instructive to describe the splitting of V1 ⊗ V2m−1 thoroughly in terms of basis vectors. To make statements less cumbersome, we let n = 2m − 1 throughout, bearing in mind that n is odd. n n−1 n−1 n Let V1 = hx, yi and Vn = ha , a b,..., ab , b i be Sp(1)-representations. The actions of sl(2, C) on V1 and Vn are given by Equations (1.14) and (1.16) of Section (1.2.1). We want to understand the actions on the tensor product

x ⊗ an, x ⊗ an−1b,..., x ⊗ abn−1, x ⊗ bn V ⊗ V = . 1 n y ⊗ an, y ⊗ an−1b,..., y ⊗ abn−1, y ⊗ bn

In particular, we would like to ﬁnd out how the left H-action interacts with the splitting ∼ V1 ⊗ Vn = Vn+1 ⊕ Vn−1. 2m ∼ Consider the structure of X2 as a K-module e : W → H ⊗ V , where H = V1 and ∼ V = Vn. Using these isomorphisms and Theorem 4.4.2, the image e(W ) is spanned by the vectors

{x⊗an, x⊗an−1b+y⊗an,..., x⊗an−kbk+y⊗an−k+1bk−1,..., x⊗bn+y⊗abn−1, y⊗bn}.

It is easier to observe the sp(1) action on the complementary subspace which we can identify as the U †-part of an SH-module. Given a suitable choice of metric, the perpendicular subspace to e(W ) is spanned by vectors of the form x⊗an−kbk −y⊗an−k+1bk−1. Calculating the action of the Casimir operator C = H2 + 2XY + 2YX reveals that

C(x ⊗ an−kbk − y ⊗ an−k+1bk−1) = (n + 1)(n − 1)(x ⊗ an−kbk − y ⊗ an−k+1bk−1).

n−k k n−k+1 k−1 This shows that x ⊗ a b − y ⊗ a b ∈ Vn−1, giving the result that

n−k k n−k+1 k−1 Vn−1 = Span{x ⊗ a b − y ⊗ a b : 1 ≤ k ≤ n}. (5.4)

The subspaces e(W ) and e(W )⊥ of the SK-module H ⊗ V are thus equivalent to the ∼ subspaces Vn+1 and Vn−1 respectively in the splitting V1 ⊗ Vn = Vn+1 ⊕ Vn−1. The SK-module structure maps σW and σV are exactly the standard real structure σn−1 and the quaternionic structure σn introduced in Section 1.2.1. The H-action is as usual determined by the action of sp(1) on x and y using the correspondence

1 ←→ x i1 ←→ ix i2 ←→ y i3 ←→ −iy.

This formalism enables us to write out the structure of V1 ⊗ Vn as an H-module in the same fashion as in Example 4.4.5.

65 AH-modules of the form V1 ⊗ V2m We can also obtain a stable AH-module from an odd-dimensional irreducible ∼ 2m+1 L M ∼ 4m+2 Sp(1)-representation V2m = C . If we take the tensor product V1 ⊗ V2m = C L M ∼ LM LM we obtain the splitting V1 ⊗ V2m = V2m+1 ⊕ V2m−1 and a left H-action in the same way as above. However this does not restrict to an H-action on any suitable real vector space 4m+2 ∼ k U such that U ⊗R C = V1 ⊗ V2m (this is obviously impossible since R 6= H for any k ). The reason for this is that the structure map σ1 ⊗ σ2m has square −1 instead of 1, L M L M and so V1 ⊗V2m is a quaternionic rather than a real representation of Sp(1) ×Sp(1) . There are two ways round this difficulty. Firstly, we could simply take the underlying 8m+4 ∼ real vector space R = V1 ⊗ V2m to be an H-module. Secondly, we can tensor with ∼ 2 H = C equipped with its standard structure map. The vector space H is unaffected L M L M by the Sp(1) × Sp(1) -action; thus we can think of V1 ⊗ V2m ⊗ H as a direct sum of L M L M two copies of V1 ⊗ V2m, which we write 2V1 ⊗ V2m. This space comes equipped with a real structure σ = σ1 ⊗ σ2m ⊗ σH , and so we have a stable AH-module L M σ LM σ ((2V1 ⊗ V2m) , (2V2m+1) ). (5.5) (As usual, once the correct structure maps have been specified, we will not usually mention the σ-superscript.) This approach is the equivalent of dealing with SK-modules 2m+1 of the form Xk=2,3 ⊗ H and the σ-sheaves O(2m + 1) ⊗ H. Both these approaches give exactly the same AH-module; both effectively leave the Sp(1) × Sp(1)-representation V1 ⊗ V2m untouched, whilst doubling the dimension of the real vector space we are considering so that it is divisible by four.

5.1.4 General Stable and Antistable AH-modules

L M LM Deﬁnition 5.1.3 Let U2n denote the AH-module (V1 ⊗ V2n+1,V2n+2). L M LM Let U2n−1 denote the AH-module (2V1 ⊗ V2n , 2V2n+1). 2n+2 The AH-module U2n corresponds to the SK-module X2 and the σ-sheaf O(2n). 2n+1 The AH-module U2n−1 corresponds to the SK-module X2 ⊗ H and the σ-sheaf O(2n − 1) ⊗ H. By analogy with the quaternions themselves, we will refer to the action of the ‘left’ L subgroup Sp(1) on V1 as the left H-action, and the action of the ‘right’ subgroup M M Sp(1) on Vn as the Sp(1) -action. (This could also be taken to signify ‘module’ L M action.) We will often omit the superscripts L and M from expressions like V1 ⊗ Vn if the context leaves no ambiguity as to which group acts on what. Thus we write

Un = aV1 ⊗ Vn+1, where a = 1 if n is even and a = 2 if n is odd. This formulation allows us to see the relationship between stable AH- and SH-modules very explicitly. The AH-module Un = aV1 ⊗ Vn+1 splits as a(Vn+2 ⊕ Vn). We can choose to regard this as a stable AH-module by thinking of aVn+2 as the ‘primed part’, or as a stable SH-module by regarding aVn as the ‘generating real subspace’. The classiﬁcation results of Sections 4.4 and 4.5 allow us to state the following theorem:

66 Theorem 5.1.4 Every stable AH-module can be written as a direct sum of the irreducibles Un with unique multiplicities.

Ln Consider the direct sum U = j=0 ajUj. We can write U more explicitly in terms of Sp(1)L × Sp(1)M -representations, as the sum

m n ! L M M M M U = V1 ⊗ V2j+1 ⊕ 2 V2k . (5.6) j=1 k=1

L M The left H-action on V1 is common to all the irreducibles. The Sp(1) -action can be much more complicated. However, we know that since Sp(1) is a compact group, any such representation can be written as a sum of irreducibles with unique multiplicities. Having done this, it is then easy to separate these representations to form separate AH- M modules, provided that each odd-dimensional representation V2k appears with even multiplicity. Thus the following is equivalent to Theorem 5.1.4:

Theorem 5.1.5 To every stable AH-module (U, U 0) can be attached an Sp(1)M -action which intertwines with the left H-action in such a way that the diagonal Sp(1)LM -action preserves U 0.

In the decomposition of Equation (5.6), each irreducible subrepresentation of the Sp(1)M -action contributes 1 to the virtual dimension of U. Thus the virtual dimension Ln P P of j=0 cjUj is equal to j even cj + 2 j odd cj.

Antistable AH-modules × Let Un = aV1 ⊗ Vn+1 be a stable AH-module. Then its dual AH-module Un is an antistable AH-module. Just like stable SH-modules, antistable AH-modules are formed by taking the smaller summand in the splitting aV1 ⊗Vn+1 = a(Vn+2 ⊕Vn). The following Lemma then follows immediately from Deﬁnition 5.1.3.

× L M LM Lemma 5.1.6 The antistable AH-module U2n takes the form (V1 ⊗ V2n+1,V2n ).

× L M LM The antistable AH-module U2n−1 takes the form (2V1 ⊗ V2n , 2V2n−1).

× 2n+2 The AH-module U2n corresponds to the SK-module X3 and the σ-sheaf O(−2n− × 2n+1 4). The AH-module U2n−1 corresponds to the SK-module X3 ⊗ H and the σ-sheaf O(−2n − 3) ⊗ H. There is a ‘unique factorisation theorem’ for antistable AH-modules which is exactly dual to Theorem 5.1.4.

5.1.5 Line Bundles over CP1 and Sp(1)-representations There is naturally a link between Sp(1)-representations and the cohomology groups of 1 0 ∼ n vector bundles over CP . In Section 4.5.1 we demonstrated that H (O(n)) = S (H), ∼ 2 where H = C is the basic representation of GL(2,C). From the inclusion SL(2,C) ⊂ GL(2,C), H is also the basic representation V1 of SL(2,C) and therefore Sp(1). The

67 n induced action of Sp(1) on S (H) is by deﬁnition the irreducible representation Vn. 1 Thus the cohomology groups of line bundles over CP are Sp(1)-representations; we have 0 ∼ 1 ∼ H (O(n)) = Vn and H (O(−n)) = Vn−2. (5.7) The exact sequence (4.11) of Section 4.5.1 is thus the same as the exact sequence ∼ 0 −→ Vn−1 −→ V1 ⊗ Vn = Vn−1 ⊕ Vn+1 −→ Vn+1 −→ 0. (5.8)

1 The fact that the cohomology groups of line bundles over CP have the structure of irreducible Sp(1)-representations is already known in the context of the theory of homogeneous spaces. Let G be a compact Lie group and let T be a maximal toral subgroup. Then the homogeneous space G/T has a homogeneous complex structure. (This famous result is due to Borel.) The right action of T on G gives G the structure of a principal T -bundle over G/T . Let t be the Lie algebra of T , so that t is a Cartan subalgebra of g. For each dominant weight λ ∈ t∗ there is a one-dimensional representation Cλ of T . The holomorphic line bundle associated to the principal bundle G and the representation λ is then

Lλ = G ×T Cλ −1 = (G × Cλ)/{(g, v) ∼ (gt, t v), t ∈ T }.

Since G acts on Lλ, the cohomology groups of Lλ are naturally representations of G. For more information see [FH, p. 382-393]. In the case of the group Sp(1), each maximal torus is isomorphic to U(1), and the ∼ 1 1 3 2 homogeneous space Sp(1)/U(1) = CP is the Hopf ﬁbration S ,→ S → S . The line −λ 1 bundle Sp(1) ×U(1) Cλ is then L , where L is the hyperplane section bundle of CP .

5.2 Sp(1)-Representations and the Quaternionic Ten- sor Product

This section describes the quaternionic algebra of stable and antistable AH-modules using the ideas of the previous section.

5.2.1 The inclusion map ιU (U)

We begin by discussing the map ιU and its image. Let Un be an irreducible stable AH-module. Then L M 0 LM † ∼ † ∗ LM Un = a(V1 ⊗ Vn+1),Un = aVn+2 and Un = (Un) = aVn .

† ∗ There is an injective map ιUn : Un → H ⊗ (Un) . This map has a natural interpretation in terms of the Sp(1)-representations involved. Writing the quaternions as the stable L R AH-module V1 ⊗ V1 , we have † ∗ ∼ L R N H ⊗ (Un) = V1 ⊗ V1 ⊗ aVn .

68 This is exactly like the motivating example of H ⊗ Ek,r in Section 3.4. Leaving the left-action untouched and taking the diagonal Sp(1)RN -action gives the isomorphism

† ∗ ∼ L RN RN H ⊗ (Un) = V1 ⊗ a(Vn+1 ⊕ Vn−1 ) (5.9)

L as an Sp(1) × Sp(1)-representation. The AH-submodule ιUn (Un) is clearly the V1 ⊗ RN † ∗ aVn+1 subrepresentation of H ⊗ (Un) . × Antistable AH-modules behave in a similar fashion. Consider the AH-module Un , so that

× L M × 0 LM × † ∼ × † ∗ LM Un = a(V1 ⊗ Vn+1), (Un ) = aVn and (Un ) = ((Un ) ) = aVn+2 . There is a similar splitting

× † ∗ ∼ L R N ∼ L RN RN H ⊗ ((Un ) ) = V1 ⊗ V1 ⊗ aVn+2 = V1 ⊗ a(Vn+3 ⊕ Vn+1 ).

× L RN This time, the A -submodule ι × (U ) is the smaller A -submodule V ⊗aV . Thus H Un n H 1 n+1 the splitting N ∼ L RN RN H ⊗ aVn = aV1 ⊗ (Vn+1 ⊕ Vn−1 ) (5.10) splits H ⊗ aVn into the direct sum of a stable AH-module isomorphic to Un and an × antistable AH-module isomorphic to Un−2. 0 × 0 The subspaces ι (U ) and ι × ((U ) ) have a similar interpretation. Treating the Un n Un n LR imaginary quaternions I as a copy of V2 , we have † ∗ ∼ LR N ∼ LRN LRN LRN I ⊗ (Un) = V2 ⊗ aVn = a(Vn+2 ⊕ Vn ⊕ Vn−2 ).

0 † ∗ and ιUn (Un) is the aVn+2 subrepresentation of I ⊗ (Un) . In exactly the same way, for × Un we have × † ∗ ∼ LR N ∼ LRN LRN LRN I ⊗ ((Un ) ) = V2 ⊗ aVn+2 = a(Vn+4 ⊕ Vn+2 ⊕ Vn ).

× 0 LRN In this case, ι × ((U ) ) is the smallest subrepresentation aV . This also shows why Un n n we would not expect U 0 to be closed under the left H-action — the group Sp(1)L does L not act upon it, since we do not have an intact copy of V1 . It is worth noting that so far we have been able consistently to interpret stable AH- modules and their subspaces as representations of highest weight in tensor products of Sp(1)-representations, and antistable AH-modules and their subspaces as representations of lowest weight.

5.2.2 Tensor products of stable AH-modules We shall now see how to use our description of stable AH-modules to form the quaternionic tensor product. The results in this section can be obtained through Quillen’s sheaf-theoretic version of AH-modules by using Theorem 4.5.7. However, the author hopes that including a little more description will help the reader to get more of a feel for what is going on. Let Um = aV1 ⊗ Vm+1, Un = bV1 ⊗ Vn+1 be stable AH-modules. By Deﬁnition 4.1.4,

† ∗ † ∗ † ∗ † ∗ Um⊗HUn = (ιUm (Um) ⊗ (Un) ) ∩ ((Um) ⊗ ιUn (Un)) ⊂ H ⊗ (Um) ⊗ (Un) .

69 In terms of Sp(1)-representations,

† ∗ † ∗ ∼ L R P Q H ⊗ (Um) ⊗ (Un) = V1 ⊗ V1 ⊗ aVm ⊗ bVn . (5.11)

∼ L RP L RP RP ∼ Using Equation (5.9), we write ιUm (Um) = aV1 ⊗ Vm+1 ⊂ V1 ⊗ a(Vm+1 ⊕ Vm−1) = † ∗ † ∗ ∼ Q H ⊗ (Um) . Tensoring this expression with (Un) = bVn gives † ∗ ∼ L RP Q ιUm (Um) ⊗ (Un) = a(V1 ⊗ Vm+1) ⊗ bVn . (5.12)

In the same way, we form the isomorphism

† ∗ ∼ P L RQ (Um) ⊗ ιUn (Un) = aVm ⊗ b(V1 ⊗ Vn+1). (5.13)

L RP Q A rearrangement of the factors leaves us considering the spaces abV1 ⊗ Vm+1 ⊗ Vn L P RQ L RP Q and abV1 ⊗ Vm ⊗ Vn+1. We now have an Sp(1) × Sp(1) × Sp(1) -representation and an Sp(1)L × Sp(1)P × Sp(1)RQ-representation. From these we want to obtain a single Sp(1) × Sp(1)-representation which leaves the left H-action intact. The way to L proceed is to leave the V1 -factor in each of these expressions alone and consider the representations of the diagonal subgroup Sp(1)RPQ. We examine the factors RP Q P RQ Vm+1 ⊗ Vn and Vm ⊗ Vn+1. To obtain a stable AH-module, we want to reduce these two Sp(1) × Sp(1)-representations to a single Sp(1)-representation. In so doing, we hope to ﬁnd the intersection of these two spaces. This is summed up in the following diagram:

† ∗ † ∗ ∼ L R P Q H ⊗ (Um) ⊗ (Un) = V1 ⊗ V1 ⊗ aVm ⊗ bVn @I @ @ † ∗ † ∗ Um ⊗ (U ) @ (U ) ⊗ Un n @ m @ ∼= @ ∼= ? @ ? † ∗ ∼ L RP Q † ∗ ∼ P L RQ ιUm (Um) ⊗ (Un) = V1 ⊗ aVm+1 ⊗ bVn (Um) ⊗ ιUn (Un) = aVm ⊗ V1 ⊗ bVn−1 @I @ @ @ @ @ @ @ ∼ L L RPQ Um⊗HUn = abV1 ⊗ ( V ? )

The upward arrows here are inclusion maps. The argument goes in the opposite direction, as we restrict our attention to particular subspaces. If we go down the left hand side, we consider the diagonal action of the subgroup Sp(1)RP , and restrict to the higher weight † ∗ RPQ subspace ιUm (Um) ⊗ (Un) . We then consider the action of Sp(1) on this. If on the other hand we go down the right hand side, we consider the diagonal action of the

70 RQ † ∗ subgroup Sp(1) , restrict to the higher weight subspace (Um) ⊗ ιUn (Un), and then consider the action of Sp(1)RPQ on this. At each ‘half-way stage’ we are considering representations of diagonal subgroups of different pairs of groups, and in both cases we take the higher weight representation in a sum Vk+1 ⊕ Vk−1 and discard the Vk−1 part. Since we do this for different diagonal subgroups we expect to be left with different subspaces. Using the Clebsch-Gordon formula, we obtain the two decompositions

min{m+1,n} min{m,n+1} RP Q ∼ M RPQ P RQ ∼ M PRQ Vm+1 ⊗ Vn = Vm+1+n−2j and Vm ⊗ Vn+1 = Vm+n+1−2j. j=0 j=0

These decompositions contain fairly similar summands, with diﬀerences arising as the index j approaches the region of min{m, n}. However, just because we have two Sp(1)RPQ-representations of the same weight, we cannot say that they are automatically the same subspace of H ⊗ (U †)∗ ⊗ (V †)∗. We want to know which parts end up contributing to the ﬁnal Sp(1)RPQ-representation whichever path we take. This will † ∗ † ∗ identify the subspace Um⊗HUn ⊆ H ⊗ (Um) ⊗ (Un) . One thing that we can guarantee for any m, n > 0 is that the representation with highest weight will be the same in both cases — both expressions have leading summand

Vm+n+1. We conjecture that this is the summand which we ﬁnd in Um⊗HUn. This would ﬁt well with the observation that stable AH-modules arise as representations of highest weight in decompositions of tensor products of Sp(1)-representations. We will show that this is in fact the case, using Joyce’s dimension formulae for stable AH-modules. Here is the main result of this section:

Theorem 5.2.1 Let Um, Un be irreducible stable AH-modules. If m or n is even then ∼ Um⊗HUn = Um+n. If m and n are both odd then ∼ Um⊗HUn = 4Um+n.

Proof. We have already noted that each irreducible representation of the Sp(1)M -action on a stable AH-module U contributes 1 to the virtual dimension of U. Thus any stable AH-module of virtual dimension k must be a sum of at least k/2 and at most k irreducibles, depending on whether the irreducibles are odd or even. We will deal with the three possible cases in turn. Case 1( m and n both even): Let m = 2p, n = 2q. Then

0 0 dim Um = 4(p + 1) dim Um = 2p + 3 dim Un = 4(q + 1) and dim Un = 2q + 3.

Using Theorem 4.1.14 we ﬁnd that dim Um⊗HUn = 4(p + q + 1) and that the virtual dimension of Um⊗HUn is equal to 1. But any stable AH-module whose virtual dimension is equal to 1 must be irreducible. The irreducible stable AH-module whose dimension

71 is 4(p + q + 1) and whose virtual dimension is 1 is V1 ⊗ V2(p+q)+1 = Um+n. Hence

Um⊗HUn = Um+n. Case 2( m even and n odd): Let m = 2p, n = 2q − 1. Then

0 0 dim Um = 4(p + 1) dim Um = 2p + 3 dim Un = 4(2q + 1) and dim Un = 4(q + 1).

Using Theorem 4.1.14 we ﬁnd that dim Um⊗HUn = 4(2p + 2q + 1) and that the virtual dimension of Um⊗HUn is equal to 2. Thus Um⊗HUn must be either an even irreducible or a sum of two odd irreducibles. † ∗ ∼ L RP Q Consider the space ιUm (Um) ⊗ (Un) = a(V1 ⊗ Vm+1) ⊗ bVn of Equation (5.12). For m = 2p and n = 2q − 1 this becomes

L RP Q ∼ L RPQ RPQ 2V1 ⊗ V2p+1 ⊗ V2q−1 = V1 ⊗ 2(V2p+2q ⊕ V2p+2q−2 ⊕ ...) (5.14)

The virtual dimension of the tensor product Um⊗HUn must be equal to 2, so we cannot have more than 2 of the irreducibles of the Sp(1)RPQ-action. We also need a total dimension of 4(2p + 2q + 1). Examining Equation (5.14) we see that the only way this ∼ RPQ can occur is if Um⊗ Un = V1 ⊗ 2V2p+2q, as all the other irreducibles of the Sp(1) - H ∼ action have smaller dimension. Hence Um⊗HUn = 2V1 ⊗ V2p+2q = Um+n. Case 3 ( m and n both odd): The argument is very similar to that of Case 2. Let m = 2p − 1, n = 2q − 1. Then

0 0 dim Um = 4(2p+1) dim Um = 4(p+1) dim Un = 4(2q +1) and dim Un = 4(q +1).

Using Theorem 4.1.14 we ﬁnd that dim Um⊗HUn = 16(p + q) and that the virtual dimension of Um⊗HUn is equal to 4. † ∗ ∼ L RP Q Consider the space ιUm (Um) ⊗ (Un) = a(V1 ⊗ Vm+1) ⊗ bVn of Equation (5.12). For m = 2p − 1 and n = 2q − 1 this becomes

L RP Q ∼ L RPQ RPQ 4V1 ⊗ V2p ⊗ V2q−1 = V1 ⊗ 4(V2p+2q−1 ⊕ V2p+2q−3 + ...). (5.15)

The only way Um⊗ Un can have a virtual dimension of four and a total dimension of H ∼ 16(p + q) is if Um⊗HUn = 4V1 ⊗ V2p+2q−1 = 4Um+n. Quaternionic tensor products of more general stable AH-modules can be computed from this result by splitting into irreducibles and using the fact that the quaternionic tensor product is distributive for direct sums. This result is parallel to Theorem 4.5.7 applied to non-negative vector bundles. For 1 0 ∼ the canonical sheaves O(n) over CP , H (O(n)) = Vn. The isomorphism O(n) ⊗O O(m) ∼= O(n + m) induces a map of cohomology groups H0(O(m)) ⊗ H0(O(n)) → H0(O(m + n)). In terms of Sp(1)-representations, this is a map

P Q ∼ PQ PQ Vm ⊗ Vn = Vm+n ⊕ Vm+n−2 ⊕ ... → Vm+n.

The map in question is projection onto the irreducible of highest weight Vn+m. This is really what this whole section has been about — the idea that the behaviour of stable AH-modules can be thoroughly and ﬂexibly described by taking subrepresentations of highest weight in tensor products of Sp(1)-representations.

72 5.2.3 Tensor Products of Antistable AH-modules It is not diﬃcult to extend Joyce’s results for tensor products of stable AH-modules to irreducible antistable AH-modules — we can follow the same argument as in the proof of Theorem 4.1.14, since the generic properties of sums and intersections guaranteed by stability also hold if one or both of the AH-modules is irreducible and antistable. Let U and V be antistable AH-modules with dim U = 4j, dim U 0 = 2j−r, dim V = 0 † ∗ † ∗ 4k and dim V = 2k − s. Let A = ιU (U) ⊗ (V ) and let B = (U ) ⊗ ιV (V ). Then dim A = 4j(2k + s) and dim B = 4k(2j + r), so dim(H ⊗ (U †)∗ ⊗ (V †)∗) = 4(2j + r)(2k + s) > dim A + dim B. Thus in generic situations we would expect dim A ∩ B = dim U⊗HV = 0. Suppose instead that V is stable, so now dim V 0 = 2k + s. A similar calculation shows that dim A + dim B ≥ dim H ⊗ (U †)∗ ⊗ (V †)∗ if and only if s(j + r) ≥ kr. In † ∗ † ∗ this case we might expect dim U⊗HV = dim A + dim B − dim(H ⊗ (U ) ⊗ (V ) ). For × × example, let U = Um = (aV1 ⊗ Vm+1) and V = Un = bV1 ⊗ Vn+1. Then we would × expect that dim(Um⊗HUn) = 2ab(m − n + 2). As in Section 5.2, we can describe what is going on in terms of diagonal actions × on tensor products of Sp(1)-representations. We will illustrate the case Um⊗HUn . Let × × † ∗ ∼ × † ∗ ∼ Um = aV1 ⊗ Vm+1 and Un = (bV1 ⊗ Vn+1) , so (Um) = Vm and ((Un ) ) = Vn+2. This gives rise to the standard descriptions † ∗ ∼ L R P ∼ L RP RP H ⊗R (Um) = aV1 ⊗ V1 ⊗ Vm = aV1 ⊗ (Vm+1 ⊕ Vm−1) and × † ∗ ∼ L R Q ∼ L RQ RQ H ⊗R ((Un ) ) = bV1 ⊗ V1 ⊗ Vn+2 = bV1 ⊗ (Vn+3 ⊕ Vn+1). The only diﬀerence between this and equation (5.9) is that we have an antistable AH- × module involved, and thus to obtain ι × (U ), we take the smaller summand of V ⊕ Um m n+3 Vn+1. We use the Clebsch-Gordon formula to describe

min{m+1,n+2}  × † ∗ ∼ L RP Q ∼ L M RPQ ιUm (Um) ⊗R ((Un ) ) = abV1 ⊗ Vm+1 ⊗ Vn+2 = abV1 ⊗  Vm+n+3−2j j=0 and min{m,n+1}  † ∗ × L P RQ L M RPQ (U ) ⊗ ι × (U ) ∼ abV ⊗ V ⊗ V ∼ abV ⊗ V . m R Un n = 1 m n+1 = 1  m+n+1−2j j=0 As with stable AH-modules, our task is to ﬁnd which of these summands is in the × × † ∗ † ∗ × intersection U ⊗ U = ι (U ) ⊗ ((U ) ) ∩ (U ) ⊗ ι × (U ). This time since m H n Um m R n m R Un n min{m + 1, n + 2} = min{m, n + 1}, we can guarantee that the summand of smallest weight will appear in both expressions. From our dimensional arguments, we only expect a non-zero intersection if m < n + 2, in which case min{m, n + 1} = m, and we would × predict that Um⊗HUn contains the summand Vn−m+1. Because its virtual dimension is × not positive, Um⊗HUn cannot be stable. This suggests that   {0} if m ≥ n + 2 × ∼ × × Um⊗HUn = (abV1 ⊗ Vn−m+1) = Un−m n or m even and m < n + 2 ×  4Un−m n and m both odd and m < n + 2. (5.16)

73 Rather than try to emulate Joyce’s (diﬃcult) proof of Theorem 4.1.14, we will conﬁrm these conjectures by appealing to Quillen’s powerful results.

× Proposition 5.2.2 Let Um be an irreducible stable AH-module and let Un be an irreducible antistable AH-module. × If m ≥ n + 2 then Um⊗HUn = {0}. × ∼ × If m < n + 2 and m or n is even then Um⊗HUn = Un−m. × ∼ × If m < n + 2 and m and n are both odd then Um⊗HUn = 4Un−m. × × × × Let Um and Un be antistable irreducible AH-modules. Then Um⊗HUn = {0}.

Proof. This follows from Theorem 4.5.7 (due to Quillen), using the correspondences + × − Um = η (aO(m)) and Un = η (aO(−n − 4)), where as usual a = 1 or 2 depending on whether m, n are even or odd. We can use this result about tensor products of antistable AH-modules to tell us about AH-morphisms between stable AH-modules, using the isomorphism ∼ × 0 HomAH(Um,Un) = (Um⊗HUn) of Theorem 4.2.9.

Proposition 5.2.3 Let Um and Un be stable AH-modules. Then (aU × )0 = aV n ≤ m Hom (U ,U ) ∼= m−n m−n AH m n {0} n > m where a = 4 if m and n are both odd and a = 1 otherwise.

Proof. This follows immediately by combining Theorems 4.2.9 and 5.2.2.

In particular, since U0 = H, we see that there are always AH-morphisms from Un into H (and indeed, this is a deﬁning property for AH-modules), but never AH-morphisms from H into Un unless n = 0. Similarly, we can now see that there are always AH-morphisms from antistable AH- modules into stable AH-modules, but never AH-morphisms from stable AH-modules into antistable AH-modules.

5.3 Semistable AH-modules and Sp(1)-representations

In this section we shall consider how the AH-module Xq ﬁts into the picture of stable AH-modules and Sp(1)-representations. Recall from Section 4.1.3 that for q ∈ S2, 0 † ∼ † ∗ ∼ Xq = H and Xq = {p ∈ H : pq = −qp} so that Xq = (Xq ) = Cq. × ∼ If we consider the dual AH-module Xq = (H, Cq) we see that the left-multiplication ∼ × Lq : H → H deﬁned by Lq(p) = q · p gives an AH-isomorphism Xq = Xq . This suggests that Theorem 4.2.9 might be particularly interesting in the case of Xq. For any AH-module U, there is a canonical isomorphism ∼ × 0 HomAH(Xq,U) = (Xq ⊗HU)

74 and so × ∼ ∼ 0 ∼ × 0 HomAH(Xq ,U) = HomAH(Xq,U) = (Xq⊗HU) = (Xq ⊗HU) . × 0 0 Let φ : Xq → U be an AH-morphism. Then we need φ(1) = u ∈ U and φ(q) ∈ U . Since φ is H-linear, φ(q) = qu, and so u ∈ U 0 ∩ qU 0. It follows that ∼ 0 ∼ 0 0 HomAH(Xq,U) = (Xq⊗HU) = U ∩ qU . (5.17) As noted by Joyce (see the summary in Section 4.1.3), the second of these isomorphisms 0 0 ∼ 0 is given by the map (idU ⊗Hχq):(U⊗HXq) → (U⊗HH) = U . 0 0 ⊥ Though Xq is not itself an Sp(1)-representation, the subspaces Cq and Xq = Cq are acted on by the Cartan subgroup U (1)q ⊂ Sp(1). As we shall see, taking the tensor product of a stable AH-module with the AH-module Xq serves to restrict attention from information about Sp(1)-representations to information concerning representations of the group U(1)q and its Lie algebra u(1)q. 2 2 2 Let Q = aI1 + bI2 + cI3 ∈ sp(1) with a + b + c = 1 and let q = ai1 + bi2 + ci3 ∈ S2. Any irreducible representation of sp(1) is also a representation of the subalgebra u(1)q = hQi. The analysis of Vn as a U(1)q-representation is already familiar: it is the decomposition of Vn into weight spaces of the operator Q : Vn → Vn. If we choose an identiﬁcation sp(1) ⊗R C = sl(2, C) such that Q = iH, this is exactly the same as the decomposition of Vn into eigenspaces of H with weights {−n, −n + 2, . . . , n − 2, n}. This decomposition gives important information about the action of q on the AH-module Un = aV1 ⊗ Vn+1. This is exactly what we have done in the explicit calculations of Section 5.1.3 for the case q = i1. We deﬁne a basis hx, yi for the space V1 in such a way that Q(x) = ix and Q(y) = −iy. This also gives the left action of q ∈ Sp(1) on V1, since the actions of q and Q coincide for this representation (see Section 1.2.1). This gives the left action of q ∈ H on Un = aV1 ⊗ Vn+1. The goal of this discussion is to describe the AH-module Un⊗HXq. We do this with the aid of the following lemma. For ease of notation we work with the AH-module Un−1.

0 0 Lemma 5.3.1 The subspace Un−1 ∩ qUn−1 is given by the sum of the weight spaces of Q with highest and lowest possible weights.

Proof. Recall from Section 5.1.3 that

x ⊗ an, x ⊗ an−1b,..., x ⊗ abn−1, x ⊗ bn V ⊗ V = , 1 n y ⊗ an, y ⊗ an−1b,..., y ⊗ abn−1, y ⊗ bn

0 ∼ and that the space (Un−1) = Vn+1 is spanned by the vectors

{x⊗an, x⊗an−1b+y⊗an,..., x⊗an−kbk+y⊗an−k+1bk−1,..., x⊗bn+y⊗abn−1, y⊗bn}.

n−k k n−k+1 k−1 n Deﬁne the basis vectors wk ≡ x ⊗ a b + y ⊗ a b , including w0 = x ⊗ a and n wn+1 = y ⊗ b . The action of H ∈ sl(2, C) is given by H(x ⊗ an−kbk) = (n − 2k + 1)x ⊗ an−kbk n−k k n−k k and H(y ⊗ a b ) = (n − 2k − 1)y ⊗ a b . Thus each basis vector wk is a weight vector with weight (n − 2k + 1). The two extreme vectors x ⊗ an and y ⊗ bn are also

75 weight vectors with weights n + 1 and −n − 1 respectively. Note that all these weights are diﬀerent. The left H-action of q on V1 ⊗ Vn is given by q(x ⊗ an−kbk) = ix ⊗ an−kbk q(y ⊗ an−kbk) = −iy ⊗ an−kbk. Left multiplication by q therefore preserves the weight space decomposition with respect to Q of a vector w ∈ V1 ⊗ Vn. n−k k n−k+1 k−1 On the basis vectors wk = x ⊗ a b + y ⊗ a b we have q(x ⊗ an−kbk + y ⊗ an−k+1bk−1) = ix ⊗ an−kbk − iy ⊗ an−k+1bk−1.

Since each vector wk has a diﬀerent weight from all the others we have q(wk) ∈ Vn+1 if and only if q(wk) ∈ hwki, and X q( λkwk) ∈ Vn+1 ⇐⇒ λk = 0 or q(wk) ∈ Vn+1 for all k. It is evident that

n n n n q(x ⊗ a ) = ix ⊗ a ∈ Vn+1 and q(y ⊗ a ) = −iy ⊗ a ∈ Vn+1, but for all the other basis vectors wk,

q(wk) 6∈ hwki and so q(wk) 6∈ Vn+1. Hence n n Vn+1 ∩ qVn+1 = hx ⊗ a , y ⊗ b i, the weight spaces with highest and lowest weight. Taking the σ-invariant subspace n n hx ⊗ a − y ⊗ b i if Un−1 is an even AH-module yields the desired result for the AH- module Un−1.

Since the AH-submodule (id ⊗Hχq)(Un−1⊗HXq) ⊂ Un−1 is the subspace generated over H by U 0 ∩ qU 0, this demonstrates the main result of this section which describes Un⊗HXq as follows:

Theorem 5.3.2 Let Un = aV1⊗Vn+1 be an irreducible stable AH-module, whose eigenspace decomposition with respect to Q ∈ sp(1) takes the form x ⊗ an+1, x ⊗ anb,..., x ⊗ abn, x ⊗ bn+1 V ⊗ V = . 1 n+1 y ⊗ an+1, y ⊗ anb,..., y ⊗ abn, y ⊗ bn+1 ∼ If Un is an even AH-module then Un⊗HXq = Xq and n+1 n+1 (id ⊗Hχq)(Un⊗HXq) = H · hx ⊗ a − y ⊗ b iR,

where the H-action is induced by the H-action on V1. ∼ If Un is an odd AH-module then Un⊗HXq = 2Xq and

n+1 n+1 n+1 n+1 (id ⊗Hχq)(Un⊗HXq) = H · hx ⊗ a , y ⊗ b iR = V1 ⊗ ha , b i.

Thus the quaternionic tensor product Un⊗HXq picks out the representations of extreme weight in the decomposition of Un into weight spaces of Q.

76 5.4 Examples and Summary of AH-modules By now, the reader should be familiar with the ideas of quaternionic algebra. In this section we shall brieﬂy sum up this information, giving explicit constructions of the AH- modules which will occur most frequently in the following chapter, in the forms in which they occur most naturally.

Example 5.4.1 Recall the AH-module Y = {(q1, q2, q3): q1i1 + q2i2 + q3i3 = 0} of Example 4.1.2. Since Y is stable and has virtual dimension 1, it follows that Y is † ∗ ∼ irreducible and is isomorphic to U2 = V1 ⊗ V3. A calculation shows that (Y ) = V2 and that the equation q1i1 + q2i2 + q3i3 = 0 is precisely the condition for (q1, q2, q3) to lie in L RM M ∼ L R M ∼ L RM RM the subspace V1 ⊗ V3 of H ⊗ V2 = V1 ⊗ V1 ⊗ V2 = V1 ⊗ (V3 ⊕ V1 ). Consider the A -modules Nk Y , Sk Y and Λk Y . Using the dimension formulae of H H H H Theorem 4.1.14 and Proposition 4.1.16, we discover that Λn Y = {0} and that Nn Y = H H SnY with dim(SnY ) = 4(n + 1) and dim(SnY )0 = 2n + 3. From this we deduce that H H H SnY ∼= U , and that all the even irreducible stable A -modules can be realised as tensor H 2n H powers of the AH-module Y .

4 Example 5.4.2 [J1, Example 10.1] Let Z ⊂ H ⊗ R be the set

Z = {(q0, q1, q2, q3): q0 + q1i1 + q2i2 + q3i3 = 0}.

∼ 3 0 Then Z = H is a left H-module. Deﬁne a real subspace Z = {(q0, q1, q2, q3): qj ∈ 0 I and q0 + q1i1 + q2i2 + q3i3 = 0}. Then dim Z = 12, dim Z = 8 and Z is a stable AH-module. ∼ In fact, Z is isomorphic to the ﬁrst odd irreducible AH-module U1 = 2V1 ⊗ V2. We † ∗ ∼ 4 ∼ have (Z ) = R = 2V1. The equation q0 + q1i1 + q2i2 + q3i3 = 0 is the condition for L RM M ∼ L R M ∼ (q0, q1, q2, q3) to lie in the subspace 2V1 ⊗ V2 of H ⊗ 2V1 = V1 ⊗ V1 ⊗ 2V1 = L RM RM 2V1 ⊗ (V2 ⊕ V0 ).

Example 5.4.3 We can obtain the rest of the odd AH-modules as tensor products of those above. From Theorem 5.2.1 we know that

U2n+1 = U2n⊗HU1. This combined with the previous examples shows that

U ∼= SnY ⊗ Z. 2n+1 H H

Thus we have obtained all the irreducible stable AH-modules in terms of previously known AH-modules. We can also use these constructions to write down formulae giving all those irreducible antistable AH-modules which are dual to stable AH-modules.

Example 5.4.4 Consider the AH-module (U, U 0) where

2 0 U = H and U = h(1, 0), (i1, i2), (0, 1), (0, i1)i.

Then {(0, q): q ∈ H} is an AH-submodule of U isomorphic to Xi1 . However, there is no ∼ complementary AH-submodule V such that U = Xi1 ⊕ V , and indeed U is irreducible.

77 It is also clear that U is not semistable — nor is it the dual AH-module of any semistable AH-module. We call such an AH-module irregular. n,α n,−α¯−1 Irregular AH-modules correspond to type (i) SK-modules of the form X1 ⊕X1 n and torsion sheaves of the form O/mx, where n ≥ 2. In the present case which singles out the subﬁeld Ci1 , we have ∼ + 2 U = η (O/m{0,∞}). 1 For a general formula linking pairs of antipodal points {z, σ(z)} in CP and complex subﬁelds of H see [Q, §14]. These are the only AH-modules which we do not describe in detail, for two reasons. Firstly, they are badly behaved compared to semistable and antistable AH-modules. Sec- ondly, as far as the author is aware, they do not arise naturally in geometrical situations in the way that the other AH-modules do.

Example 5.4.5 There is one remaining irreducible to consider — the AH-module (H, {0}). This trivially satisﬁes Deﬁnition 4.1.1, and so is an AH-module. Any AH- module whose primed part is zero is a direct sum of copies of (H, {0}). It is easy to see that for any irreducible AH-module U, U⊗H(H, {0}) = {0} unless U = H, in which case we have H⊗H(H, {0}) = (H, {0}). Though badly behaved, the AH-module (H, {0}) does arise naturally in quaternionic × algebra — for example, Z⊗HH = (H, {0}). This suggests the notation

× (H, {0}) = U−1, in which case this result agrees with Theorem 5.2.2. Such notation is consistent with the sheaf description of antistable AH-modules, since we have

× − U−1 = η (2O(−3)) as expected. Thus we interpret (H, {0}) as the ‘antistable part’ of 2V1 ⊗ V0. The dual space (H, H) is of course not an AH-module, so there is no stable AH-module U−1 which × × is dual to U−1. In spite of this we still regard U−1 as ‘antistable’, because treating it as an exception every time would be cumbersome. We end this chapter with a diagram (overleaf) summarising much of our theory.

78 Figure 5.1: Irreducible AH-modules and the ratio of their dimensions.

AH-morphisms -

Xq and irregular × ∼ × AH-modules ∼ Y = U2 Y = U2

× ∼ ∼ Z = U1 Z = U1

× U−1 = (H, {0}) × H ? ? ? ? ? H ...... 0 1 1 3 U 0/ dim U 1dim 4 2 @I 4 @ @ × Un (3 ≤ n < ∞) Un (3 ≤ n < ∞) Quaternionic tensor products

This describes the AH-modules we have met so far — all the ﬁnite-dimensional irreducible AH-modules. The quaternionic tensor product of two AH-modules is always to the left of both of them in Figure 5.1. On the other hand, there are AH-morphisms from an AH-module into itself and any AH-modules to its right — and never from an AH- module to any AH-module to its left. These statements are closely linked by Theorem 4.2.9. If U and V are irreducible stable or antistable AH-modules, this demonstrates that that there will always be AH-morphisms from U⊗HV into U and V , but never any AH-morphisms from U or V into U⊗HV , unless U or V is equal to H.

79 Chapter 6

Hypercomplex Manifolds

This chapter uses the quaternionic algebra developed in Chapters 4 and 5 to describe hypercomplex manifolds. It is in three parts (the first of which is a summary of Joyce’s work, the other two being original). The first part (Section 6.1) summarises Joyce’s theory of q-holomorphic functions. A q-holomorphic function on a hypercomplex manifold M is a smooth function f : M → H which satisfies a quaternionic version of the Cauchy-Riemann equations. We let PM denote the AH-module of q-holomorphic functions on M. The q-holomorphic functions on the hypercomplex manifold H are precisely the regular functions of Fueter and Sudbery. Because of the noncommuta- tivity of the quaternions, the product of two q-holomorphic functions is not in general q-holomorphic. Nonetheless, q-holomorphic functions possess a rich algebraic structure. Joyce attempts to capture this using the concept of an H-algebra, a quaternionic version of a commutative algebra over the real or complex numbers. ∗ ∼ In the second part (Section 6.2), we use the Sp(1)-representation T M = 2nV1 defined by the hypercomplex structure to obtain a natural splitting of the quaternionic ∗ ∼ cotangent space of a hypercomplex manifold M, which we write H ⊗ T M = A ⊕ B. ∼ This is precisely a version of the splitting H ⊗ aVk = aV1 ⊗ (Vk+1 ⊕ Vk−1) used in the previous chapter to construct and describe all stable and antistable AH-modules, and the Sp(1)-version of quaternionic algebra gives a complete description of the geometric situation. A function f is q-holomorphic if and only if its differential df takes values in the subspace A ⊂ H ⊗ T ∗M. This is very similar to the situation in complex geometry where the differential of a holomorphic function takes values in the holomorphic cotangent space Λ1,0. It follows that the subbundle A should be thought of as the q-holomorphic cotangent space of M, its complement B being the q-antiholomorphic cotangent space. The third part (Sections 6.3 and 6.4) is about AH-bundles, which are smooth vector bundles whose fibres are AH-modules. They can be defined over any smooth manifold M, but are most interesting when M is hypercomplex. An AH-bundle E is said to be q-holomorphic if it is simultaneously holomorphic with respect to the whole 2-sphere of complex structures on M. This condition is met if and only if E carries an anti- self-dual connection which is compatible with the structure of E as an AH-bundle. We generalise the theory of q-holomorphic functions to that of q-holomorphic sections, so that a q-holomorphic function is precisely a q-holomorphic section of the trivial bundle M × H. We investigate the algebraic structure of q-holomorphic sections in some detail, showing that the q-holomorphic sections of a q-holomorphic vector bundle E form an

80 H-algebra module over PM .

6.1 Q-holomorphic Functions and H-algebras

6.1.1 Quaternion-valued functions This section is mainly a summary of [J1, §§3,5], to which the reader is referred for more detail and proofs of the important results. Let M be a smooth manifold and let C∞(M, H) be the vector space of smooth quaternion-valued functions on M. An H-action on C∞(M, H) is deﬁned by setting (q · f)(m) = q(f(m)) for all m ∈ M, f ∈ C∞(M, H). Thus C∞(M, H) is a left H-module. What is less immediately obvious is that C∞(M, H) is an AH-module. Lemma 6.1.1 Deﬁne a linear subspace

∞ 0 ∞ ∞ C (M, H) = {f ∈ C (M, H): f(m) ∈ I for all m ∈ M} = C (M, I).

With these deﬁnitions, (C∞(M, H),C∞(M, H)0) is an AH-module.

Proof. We use the fact that M can be embedded in C∞(M, H)† in the following way. For each m ∈ M, deﬁne the ‘evaluation map’

∞ θm : C (M, H) → H by θm(f) = f(m).

∞ × ∞ Then θm(q · f) = q · θm(f), so θm ∈ C (M, H) . Also, if f ∈ C (M, I) then θm(f) ∈ I ∞ † for all m ∈ M, so θm ∈ C (M, H) . ∞ ∞ † Suppose that f ∈ C (M, H), and α(f) = 0 for all α ∈ C (M, H) . Since θm ∈ C∞(M, H)†, f(m) = 0 for all m ∈ M ; so f ≡ 0. Thus C∞(M, H) is an AH-module, by Deﬁnition 4.1.1. This technique of linking a point m ∈ M to an element of C∞(M, H)† is extremely useful. Joyce has used this process to reconstruct hypercomplex manifolds from their H-algebras. Note that we have assumed no geometric structure on M other than that of a smooth manifold.

Complex and quaternionic functions

2 For each q ∈ S let ιq : C → H be the inclusion obtained by identifying i ∈ C with 2 1 q ∈ S , so that ιq(a + ib) = a + qb. For any real vector space E, the inclusion ιq extends to a map ιq : C ⊗ E → H ⊗ E given by ιq(e0 + ie1) = e0 + qe1 (for e0, e1 ∈ E ). Note that the images of ιq and ι−q are the same, but the two maps are not identical: since ιq(a + ib) = a + qb and ι−q(a + ib) = a − qb, we see that ιq(z) = ι−q(¯z). Let f = a + ib be a complex-valued function on M. Then for every q ∈ S2, the function ιq(f) = a + qb is a quaternion-valued function on M. In this way we obtain quaternion-valued functions from complex ones, and we shall soon see that this construction can be used to obtain q-holomorphic functions from holomorphic ones.

1 The map ιq will be distinguished from the inclusion map ιU of AH-modules by the use of lower-case rather than upper-case subscripts.

81 6.1.2 Q-holomorphic functions In this section we will deﬁne q-holomorphic functions, the hypercomplex version of holomorphic functions, and familiarise ourselves with some of their basic properties. Consider ∞ the complex manifold (M,I). A complex-valued function f = f0 + idf1 ∈ C (M, C) is holomorphic (with respect to I ) if and only if f satisﬁes the Cauchy-Riemann equations

df0 + Idf1 = 0. (6.1)

Let (M,I1,I2,I3) be a hypercomplex manifold. Here is the definition of a q-holomorphic function on M. Definition 6.1.2 Let f ∈ C∞(M, H) be a smooth H-valued function on M. Then ∞ f = f0 +f1i1 +f2i2 +f3i3 with fj ∈ C (M). The function f is said to be q-holomorphic if and only if it satisfies the Cauchy-Riemann-Fueter equations

Df ≡ df0 + I1df1 + I2df2 + I3df3 = 0.

The set of all q-holomorphic functions on M is called PM . The term q-holomorphic is short for quaternion-holomorphic, and it is intended to indicate that a q-holomorphic function on a hypercomplex manifold is the appropriate quaternionic analogue of a holomorphic function on a complex manifold. If a function f is q-holomorphic then the function q · f is also q-holomorphic for all q ∈ H. Thus the ∞ set of q-holomorphic functions PM forms a left H-submodule of C (M, H). We adopt 0 the obvious deﬁnition for PM , namely 0 PM = {f ∈ PM : f(m) ∈ I for all m ∈ M}.

0 ∞ 0 ∞ So PM = PM ∩ C (M, H) , and PM is an AH-submodule of C (M, H). Thus the q-holomorphic functions PM on a hypercomplex manifold M form an AH-module. The product of two q-holomorphic functions is not in general q-holomorphic — we can observe this simply by noting that all constant functions are trivially q-holomorphic, but PM is not even closed under right-multiplication by quaternions. Thus q-holomorphic functions do not form an algebra in the same sense that the holomorphic functions on a complex manifold do. We show that there are many interesting q-holomorphic functions on a hypercomplex manifold M, by observing that every complex-valued function on M which is holomorphic with respect to any complex structure Q ∈ S2 gives rise to a q-holomorphic function.

2 Lemma 6.1.3 Let q = a1i1 + a2i2 + a3i3 ∈ S and let Q = a1I1 + a2I2 + a3I3 be the corresponding complex structure on M. Let f = x + iy ∈ C∞(M, C) be holomorphic with respect to Q. Then ιq(f) = x + qy is a q-holomorphic function.

Proof. The proof is a simple substitution. If f = x + iy is holomorphic with respect to Q then it satisﬁes the Cauchy-Riemann equations dx + Q(dy) = 0, in which case

0 = dx + (a1I1 + a2I2 + a3I3)(dy) = D(x + a1yi1 + a2yi2 + a3yi3) = D(x + qy).

82 So the H-valued function ιq(f) = x + qy is q-holomorphic. If f = x + iy ∈ C∞(M, C) is holomorphic with respect to Q, then its conjugate f¯ = x − iy is holomorphic with respect to −Q. Does this mean that both x + qy and x − qy are q-holomorphic functions? Closer inspection shows that this is not the case — we see that ιq(x + iy) = ι−q(x − iy) = x + qy. So the holomorphic functions with respect to Q and −Q are mapped to the same ∞ functions when we consider their images under ι±q in C (M, H). We shall see that this is a consequence of (indeed is equivalent to) the fact that the AH-modules Xq and X−q are the same. ∞ Now, if the functions f, g ∈ C (M, Cq) are both holomorphic with respect to Q, ∞ then so is fg ∈ C (M, Cq). By Lemma 6.1.3, ιq(f), ιq(g), and ιq(fg) = ιq(f)ιq(g) are all q-holomorphic. In this special case where the q-holomorphic functions f and g both take values in a commuting subfield of H, their product is also q-holomorphic. This is reminiscent of the situation described in Lemma 4.1.7, where two elements u ∈ † ∗ † ∗ U and v ∈ V such that ιU (u) ∈ Cq ⊗ (U ) and ιV (v) ∈ Cq ⊗ (V ) define an element u⊗Hv ∈ U⊗HV . In this case we have two well-defined ‘products’ of f and g — their quaternionic tensor product f⊗Hg ∈ PM ⊗HPM and their product as Cq- valued functions fg = gf ∈ PM . This algebraic situation is described by the theory of H-algebras.

6.1.3 H-algebras and Q-holomorphic functions

An H-algebra is a quaternionic version of an algebra over a commutative field. Let F be a commutative field (usually the real or complex numbers). An F-algebra is a vector space A over F, equipped with an F-bilinear multiplication map µ : A × A → A which has certain algebraic properties. For example if µ(a, b) = µ(b, a) for all a, b ∈ A then µ is said to be commutative. As we have already seen in Section 1.3, this formulation is of no great use to us when seeking a quaternionic analogue because the non-commutativity of the quaternions makes the notion of an H-bilinear map untenable. The axioms for an algebra over F can alternatively be written in terms of tensor products. The main feature of tensor algebra is that a bilinear map on the cartesian product A×B translates into a linear map on the tensor product A⊗F B. So our bilinear multiplication map µ : A × A → A becomes an F-linear map µ : A ⊗F A → A. The commutative axiom µ(a, b) = µ(b, a) becomes µ(a ⊗ b) = µ(b ⊗ a), so µ(a ⊗ b − b ⊗ a) = µ(a ∧ b) = 0. Hence we obtain a ‘tensor algebra version’ of the commutative axiom, 2 saying that µ : A ⊗F A → A is commutative if and only if Λ A ⊆ ker µ. Once we have reformulated our axioms in terms of tensor products and linear maps we can translate them into quaternionic algebra, replacing ‘vector space’, ‘linear map’ and ‘tensor product’ with ‘AH-module’, ‘AH-morphism’ and ‘quaternionic tensor product’. This is precisely what Joyce does, producing the following definition [J1, §5]:

Axiom H. (i) P is an AH-module.

(ii) There is an AH-morphism µP : P⊗HP → P, called the multiplication map.

83 (iii)Λ2 P ⊂ ker µ . Thus µ is commutative. H P P (iv) The AH-morphisms µP : P⊗HP → P and id : P → P combine to give AH-morphisms µP ⊗H id and id ⊗HµP : P⊗HP⊗HP → P⊗HP. Composing with µP gives AH-morphisms µP ◦ (µP ⊗H id) and µP ◦ (id ⊗HµP ): P⊗HP⊗HP → P. Then µP ◦ (µP ⊗H id) = µP ◦ (id ⊗HµP ). This is associativity of multiplication. (v) An element 1 ∈ A called the identity is given, with 1 ∈/ P0 and I·1 ⊆ P0. (vi) Part (v) implies that if α ∈ P† then α(1) ∈ R. Thus for each a ∈ P, 1⊗Ha and a⊗H1 ∈ P⊗HP by Lemma 4.1.7. Then µP (1⊗Ha) = µP (a⊗H1) = a for each a ∈ P. Thus 1 is a multiplicative identity.

Deﬁnition 6.1.4 P is an H-algebra if P satisﬁes Axiom H.

Here H-algebra stands for Hamilton algebra. We also define morphisms of H-algebras: Definition 6.1.5 Let P, Q be H-algebras, and let φ : P → Q be an AH-morphism. Write 1P , 1Q for the identities in P, Q respectively. We say that φ is an H-algebra morphism if φ(1P ) = 1Q and µQ ◦ (φ⊗Hφ) = φ ◦ µP as AH-morphisms P⊗HP → Q. Let M be a hypercomplex manifold. Joyce has constructed a multiplication map µM with respect to which the q-holomorphic functions PM form an H-algebra, and such ∞ that for f, g ∈ C (M, Cq) holomorphic with respect to Q we have µ(f⊗Hg) = fg. In this way the the H-algebra PM includes the algebras of holomorphic functions with respect to all the different complex structures on M. Let M, N and (therefore) M × N be hypercomplex manifolds. We define an AH- † † morphism φ : PM ⊗HPN → PM×N . Let U and V be AH-modules. Let x ∈ U ⊗ V † † and y ∈ U⊗HV . Then y · x ∈ H, where ‘ · ’ contracts together the factors of U ⊗ V † ∗ † ∗ † ∗ † ∗ with those of (U ) ⊗ (V ) in U⊗HV ⊂ H ⊗ (U ) ⊗ (V ) . Define a linear map

† † † λUV : U ⊗ V → (U⊗HV ) by setting λUV (x)(y) = y · x, (6.2) † † 2 for all x ∈ U ⊗ V and y ∈ U⊗HV . † Consider again the maps θm ∈ PM of Lemma 6.1.1, which allow us to interpret points of m ∈ M as AH-morphisms θm : PM → H. In the same way, we can associate † † to each point (m, n) ∈ M × N the map θm ⊗ θn ∈ PM ⊗ PN . Then λPM ,PN (θm ⊗ θn) ∈ † (PM ⊗HPN ) . ∞ Definition 6.1.6 We define a map φ : PM ⊗HPN → C (M × N, H) as follows. Let f ∈ PM ⊗HPN . Then λPM ,PN (θm ⊗ θn) · f ∈ H and we define φ(f): M × N → H by setting

φ(f)(m, n) = λPM ,PN (θm ⊗ θn) · f. For infinite-dimensional vector spaces U and V we define U ⊗ V to consist of finite sums of elements u ⊗ v. Thus φ(f) is a sum of finitely many smooth functions, and so is smooth. It is easy to see that φ(f) is q-holomorphic, since DM×N =

2 See [J1, Deﬁnition 4.2], where Joyce deﬁnes this map and uses it to prove that U⊗HV is an AH- module.

84 DM (φ(f)) + DN (φ(f)) = 0, i.e. we can evaluate the derivatives in the M and N 0 0 0 directions separately. It is clear that φ is H-linear and that φ(PM ⊗HPN ) ⊆ PM×N . Thus we have a canonical AH-morphism φ : PM ⊗HPN → PM×N .

The second (and easier) step is to consider the case M = N, so φ : PM ⊗HPM → PM×M . The diagonal submanifold Mdiag = {(m, m): m ∈ M} ⊂ M × M is a submanifold of M × M isomorphic to M as a hypercomplex manifold, and each q-holomorphic function on M × M restricts to a q-holomorphic function on Mdiag. Let ρ be the restriction map ρ : PM×M → PMdiag ; then ρ is an AH-morphism. Thus we can deﬁne an AH-morphism µM = ρ ◦ φ : PM ⊗HPM → PM . (6.3) Here is the key theorem of this section:

Theorem 6.1.7 [J1, 5.5]. Let M be a hypercomplex manifold, so that PM is an AH- module. Let 1 ∈ PM be the constant function on M with value 1, and let µM :

PM ⊗HPM → PM be the AH-morphism µM = ρ ◦ φ of Equation (6.3). With these definitions, PM is an H-algebra. ∞ ∞ This statement is also true for C (M, H). An AH-morphism φ : C (M, H)⊗H C∞(N, H) → C∞(M × N, H) can be defined just as in Definition 6.1.6, and ρ is still the obvious restriction. The q-holomorphic functions PM form an H-subalgebra ∞ ∞ of C (M, H), and the inclusion map id : PM → C (M, H) is an H-algebra morphism. Example 6.1.8 Q-holomorphic functions on H [J1, §10]. Consider the manifold H with coordinates (x0, x1, x2, x3) representing the quaternion x0 + x1i1 + x2i2 + x3i3. Then H is naturally a complex manifold with hypercomplex structure (I1,I2,I3) given by

I1dx2 = dx3,I2dx3 = dx1,I3dx1 = dx2 and Ijdx0 = dxj, j = 1, 2, 3. (6.4) Consider the linear H-valued functions on H. Using the Cauchy-Riemann-Fueter Equation (1.20) with the hypercomplex structure in (6.4), we ﬁnd that f = q0x0 + q1x1 + q2x2 + q3x3 is q-holomorphic if and only if q0 + q1i1 + q2i2 + q3i3 = 0. Thus the linear q-holomorphic functions on H form an AH-submodule of the set of all linear ∼ functions, and this submodule is isomorphic to the AH-module Z = U1 of Example 5.4.2. Consider the AH-module U (k) of homogeneous q-holomorphic polynomials of degree k. The spaces U (k) are important in quaternionic analysis and are studied by Sudbery [Su, §6]. Joyce uses the dimension formulae of Proposition 4.1.16 to prove that Sk Z ∼= H U (k). We can easily show that (k + 1)U k even Sk Z ∼= (k + 1)V ⊗ V = k H 1 k+1 1 2 (k + 1)Uk k odd. This constructs not only the spaces U (k), but also their structure as Sp(1)-representations, which is crucial to Sudbery’s approach. The space of all q-holomorphic polynomials on H is therefore given by the sum L∞ Sk Z, which is naturally an H-algebra. Joyce [J1, Example 5.1] calls this F Z , the k=0 H free H-algebra generated by Z , and this idea enables us to write down the multiplication map µH. The full H-algebra PH is constructed by adding in convergent power series.

85 6.2 The Quaternionic Cotangent Space

∗ 1,0 0,1 Let M be a complex manifold and recall the splitting C ⊗R T M = Λ M ⊕ Λ M of Equation (2.1). A function f ∈ C∞(M, C) is holomorphic if df ∈ Λ1,0M. The Cauchy-Riemann operator is deﬁned by ∂ = π0,1 ◦ d, and f is holomorphic if and only if ∂f = 0. This section presents the quaternionic analogue of this description. Let M be a hypercomplex manifold. We show that there is a natural splitting of the quaternionic ∗ ∼ cotangent space H ⊗ T M = A ⊕ B. The bundle A is then the q-holomorphic cotangent space of M. A function f ∈ C∞(M, H) is q-holomorphic if and only if df ∈ A, and the Cauchy-Riemann-Fueter operator can be written δ¯ = πB ◦ d, where πB is the natural projection to the subspace B ⊂ H ⊗ T ∗M. The ease with which quaternionic algebra presents such close parallels with complex geometry could scarcely be more rewarding.

Theorem 6.2.1 Let M 4n be a hypercomplex manifold and let H ⊗ T ∗M be the quaternionic cotangent bundle of M. The hypercomplex structure determines a natural splitting

∗ ∼ H ⊗ T M = A ⊕ B, where dim A = 12n and dim B = 4n.

∗ ∼ Proof. Recall from Section 3.2 that T M = 2nV1 as an Sp(1)-representation. Call this G representation 2nV1 (since it is the action deﬁned by the geometric structure of M ). Following the standard arguments of Chapter 5, we have a splitting

∗ ∼ L R G H ⊗ T M = V1 ⊗ V1 ⊗ 2nV1 ∼ L RG RG = 2nV1 ⊗ (V2 ⊕ V0 ) ∼= A ⊕ B,

L RG L RG where A = 2nV1 ⊗ V2 and B = 2nV1 ⊗ V0 . This is a situation with which we are by now very familiar. Using the theory of Chapter 5, we immediately deduce that

∼ • The (ﬁbres of the) subspaces A and B are AH-modules with A = nU1 and ∼ × B = nU−1.

• Dim A = 12n and dim A0 = 8n, where A0 = A ∩ I ⊗ T ∗M. A is a stable AH- module.

• Dim B = 4n and dim B0 = 0, where B0 = B ∩ I ⊗ T ∗M. B is an antistable AH-module.

• The mapping A ⊕ B,→ H ⊗ T ∗M is an injective AH-morphism which is an isomorphism of the total spaces. It is not an AH-isomorphism because dim(A0⊕B0) is smaller than dim(I ⊗ T ∗M).

86 The importance of this splitting lies partly in the following result:

Theorem 6.2.2 A function f ∈ C∞(M, H) is q-holomorphic if and only if df ∈ C∞(A).

The proof of this theorem will be in two parts. Firstly, we investigate the projection operators πA and πB from H ⊗ T ∗M onto A and B. We then show that the Cauchy- Riemann-Fueter operator can be written as πB ◦ d. To work out projection maps to A and B we use the Casimir operator for the diagonal Sp(1)RG-action, obtained by coupling right-multiplication on H with the action of the hypercomplex structure. The diagonal Lie algebra action is given by

I(ω) = I1(ω) − ωi1 J (ω) = I2(ω) − ωi2 K(ω) = I3(ω) − ωi3. (6.5)

2 2 2 Then C(ω) = (I +J +K )(ω) = −6ω−2χ(ω), where χ(ω) = I1(ω)i1+I2(ω)i2+I3(ω)i3. The operator χ satisﬁes the equation χ2 = 3 − 2χ, so has eigenvalues +1 and −3. If RG ω is in the −3 eigenspace then we have C(ω) = 0, so ω ∈ V0 . If ω is in the +1 RG eigenspace then we have C(ω) = −8, so ω ∈ V2 . Thus the subspaces A and B are the eigenspaces of the operator χ. This enables us to write down expressions for πA and πB such that πA + πB = 1.

Lemma 6.2.3 Projection maps πA : H⊗T ∗M → A and πB : H⊗T ∗M → B are given by 1 1 πA(ω) = (3ω+I (ω)i +I (ω)i +I (ω)i ) πB(ω) = (ω−I (ω)i −I (ω)i −I (ω)i ). 4 1 1 2 2 3 3 4 1 1 2 2 3 3 These maps depend only on the structure of M as a hypercomplex manifold.

Just as we defined the Dolbeault operators in Equation (2.4), we define new differential operators δ and δ¯:

δ : ∞(M, ) → C∞(M,A) δ¯ : ∞(M, ) → C∞(M,B) C H and C H (6.6) δ = πA ◦ d δ¯ = πB ◦ d.

Then for a function f ∈ C∞(M, H), df = δf + δf¯ .

Proposition 6.2.4 A function f ∈ C∞(M, H) is q-holomorphic if and only if δf¯ = 0.

Proof. Let f = f0 + f1i1 + f2i2 + f3i3 be a q-holomorphic function on M, so Df = df0 + I1df1 + I2df2 + I3df3 = 0. This is exactly the real part of the equation ¯ 4δf ≡ df − I1dfi1 − I2dfi2 − I3dfi3 = 0. (6.7)

Moreover, the three imaginary parts are the equations Ij(Df) = 0, which are satisﬁed if and only if Df = 0. Thus the real equation Df = 0 is exactly the same as the quaternionic equation δf¯ = 0.

87 It now follows that a function f is q-holomorphic if and only if df = δf ∈ C∞(M,A). This proves Theorem 6.2.2 and motivates the following deﬁnition: ∗ ∼ Deﬁnition 6.2.5 Let M be a hypercomplex manifold and let H ⊗ T M = A ⊕ B, as in Theorem 6.2.1. Then A is the q-holomorphic cotangent space of M and B is the q-antiholomorphic cotangent space of M.

Here are some practical methods for writing the spaces A and B. Let ω = 1 ⊗ ω0 + i1 ⊗ ω1 + i2 ⊗ ω2 + i3 ⊗ ω3. It is straightforward to verify that

ω ∈ A ⇐⇒ ω0 + I1(ω1) + I2(ω2) + I3(ω3) = 0, and ω ∈ B ⇐⇒ ω0 = I1(ω1) = I2(ω2) = I3(ω3). Let M 4n be a hypercomplex manifold and let {ea : a = 0,..., 4n − 1} be a basis ∗ 4a 4a+b for Tx M such that Ib(e ) = e . In other words, we choose a particular isomorphism ∗ ∼ n 4a 4a+1 4a+2 4a+3 Tx M = H such that the subspace he , e , e , e i is a copy of H with its standard hypercomplex structure. With respect to this basis the ﬁbres of A and B at the point x are given by

nX a o Ax = qae : q4b + q4b+1i1 + q4b+2i2 + q4b+3i3 = 0 for all b = 0, . . . , n − 1 (6.8) and

nX a o Bx = qae : q4b = −q4b+1i1 = −q4b+2i2 = −q4b+3i3 for all b = 0, . . . , n − 1 . (6.9) Equation (6.8) is a higher dimensional version of the equation q0 + q1i1 + q2i2 + q3i3 = 0 which gives the q-holomorphic cotangent space on H (Example 6.1.8). This gives an ∼ AH-isomorphism between Ax and the AH-module nZ = nU1. Let us pause for a moment to reﬂect on what has happened. On a complex manifold, we have a splitting of C ⊗ T ∗M into two spaces of equal dimension, which are conjugate to one another. On a hypercomplex manifold, we have a splitting into two subspaces, but they are not equal in size. Instead, one of them has three times the dimension of the ∼ other, because the subspaces are deﬁned by the equation V1 ⊗ V1 = V2 ⊕ V0. This type of splitting into spaces of dimension 3n and n is typical of the quaternions, echoing the ∼ fundamental isomorphism H = I ⊕ R.

6.2.1 The Holomorphic and Q-holomorphic cotangent spaces

4n Let M be a hypercomplex manifold with hypercomplex structure (I1,I2,I3). Let 2 q = a1i1 + a2i2 + a3i3 ∈ S and let Q = a1I1 + a2I2 + a3I3 be the corresponding complex structure on M. We have already seen (Lemma 6.1.3) that if f is a complex-valued function on M which is holomorphic with respect to the complex structure Q, then ιq(f) is a q-holomorphic function. This is a correspondence not only of holomorphic and q-holomorphic functions, but also of the holomorphic and q-holomorphic cotangent spaces.

88 1,0 Each complex structure Q on M deﬁnes a holomorphic cotangent space ΛQ . Using ∗ ∼ ∗ ∗ the embedding ιq we can consider C⊗T M = Cq ⊗T M as a subspace of H⊗T M. The relationship between these subspaces of H ⊗ T ∗M is clariﬁed in the following Lemma: Lemma 6.2.6 Let A ⊂ H ⊗ T ∗M be the q-holomorphic cotangent space of a hyper- 1,0 ∗ complex manifold M and let ΛQ ⊂ C ⊗ T M be the holomorphic cotangent space with respect to the complex structure Q. Then 1,0 H · ιq(ΛQ ) = (idA⊗Hχq)(A⊗HXq).

Proof. We illustrate the case dim M = 4, the general result holding in exactly the same way but involving more coordinates. For any point x ∈ M we choose a basis 0 1 2 3 ∗ {e , e , e , e } for Tx M such that the hypercomplex structure at x is given by the standard relations (6.4). Without loss of generality take q = i1 and consider the complex manifold (M,I1). Then Λ1,0 = he0 + ie1, e2 + ie3i . I1 C

Since ιi1 (a + ib) = a + i1b, we see that · ι (Λ1,0) = · he0 + i e1, e2 + i e3i. (6.10) H q I1 H 1 1 0 0 0 Recall from Section 5.3 that (idU ⊗Hχq)(U⊗HXq) = U ∩ qU for any AH-module 0 3 U. Using Equation (6.8), we have A = {(q0e + ... + q3e ): q0 + q1i1 + q2i2 + q3i3 = 0} and A0 = I ⊗ T ∗M. It follows that 0 0 0 3 A ∩ i1A = {q0e + ... + q3e ∈ A : qj ∈ hi2, i3i} 0 1 0 1 2 3 2 3 = hi2e − i3e , i3e + i2e , i2e − i3e , i3e + i2e i, (6.11) ∼ and the image (idA ⊗Hχi1 )(A⊗HXi1 ) = 2Xi1 is generated over H by this subspace. 0 1 0 1 2 3 2 3 Observe that −i2 · (i2e − i3e ) = e + i1e and that −i2 · (i2e − i3e ) = e + i1e . Therefore · (A0 ∩ i A0) is exactly the same as the subspace · ι (Λ1,0) of Equation H 1 H q I1 (6.10). This gives us a description of the q-holomorphic and q-antiholomorphic cotangent spaces of M in terms of complex geometry: Corollary 6.2.7 Let M 4n be a hypercomplex manifold, with q-holomorphic cotangent space A and q-antiholomorphic cotangent space B. Then

X 1,0 A = H · ιq(ΛQ ) Q∈S2 and \ 0,1 B = H · ιq(ΛQ ). Q∈S2 Thus the q-holomorphic cotangent space is generated over H by the diﬀerent holomorphic cotangent spaces.

Proof. Use the fact that A is a stable AH-module; thus A is generated over H by the subspaces A0 ∩ qA0 for q ∈ I. The proof now follows immediately from Lemma 6.2.6.

89 6.3 AH-bundles

An AH-bundle is the natural quaternionic analogue of a real or complex vector bundle: a set of AH-modules E parametrised smoothly by a base manifold M. Definition 6.3.1 Let M be a differentiable manifold and (W, W 0) a fixed AH- module. A smooth AH-module bundle with fibre W or simply AH-bundle is a family 0 {(Ex,Ex)}x∈M of AH-modules parametrised by M, together with a differentiable man- S ifold structure on E = x∈M Ex such that ∞ • The projection map π : E → M, π : Ex 7→ x is C .

• For every x0 ∈ M, there exists an open set U ⊆ M containing x0 and a diﬀeo- morphism −1 φU : π (U) → U × W

such that φ(Ex) 7→ {x} × W is an AH-isomorphism for each x ∈ U.

A section of an AH-bundle is a smooth map α : M → E such that α(x) ∈ Ex for all x ∈ M, and the left H-module of smooth sections of E is denoted by C∞(M,E). Define C∞(M,E)0 = C∞(M,E0). With this definition, it is easy to adapt Lemma 6.1.1 to show that if E is an AH-bundle then C∞(M,E) is an AH-module. We can multiply sections on the left by quaternion-valued functions, and C∞(M,E) is a left module over the ring C∞(M, H). The standard real or complex algebraic operations which give new bundles from old can be transfered en masse to quaternionic algebra and AH-bundles. Let (E, π1,M) and (F, π2,N) be AH-bundles. We define a bundle (E⊗HF,M×N) by setting (E⊗HF )(m,n) = Em⊗HFn. We can define a bundle E ⊕ F in the same way. In particular, let E and F be AH-bundles over M. By restricting to the diagonal submanifold of M × M we define a bundle (E⊗HF,M) by setting (E⊗HF )m = E ⊗ F . In the same way (E ⊕ F,M), Λk E and Sk (E) are A -bundles, as is E× m H m H H H if the fibres of E are SAH-modules. We will always make clear whether an AH-bundle ‘ E⊗HF ’ refers to (E⊗HF,M × M) or the diagonal restriction (E⊗HF,M). The q-holomorphic and q-antiholomorphic cotangent bundles A and B are both AH-bundles, and are AH-subbundles of H ⊗ T ∗M. In fact, every AH-bundle can be regarded as an AH-subbundle of some bundle H ⊗ V , as is easy to demonstrate:

Lemma 6.3.2 Let E = (E, π, M) be an AH-bundle. Then E is isomorphic to an AH-subbundle of H ⊗ V , for some real vector bundle (V, π1,M).

Proof. This is an extension of the fact that any AH-module W is isomorphic to ιW (W ) ⊆ † ∗ † ∗ † ∗ H ⊗ (W ) . Consider the map ιEx (Ex): Ex → H ⊗ (Ex) for x ∈ M. Let Vx = (Ex) . S Then V = x∈M Vx is a real vector bundle, and H ⊗ V is an AH-bundle. Thus E is isomorphic to the AH-subbundle [ ιEx (Ex) ⊆ H ⊗ V, x∈M and the lemma is proved.

90 Abusing notation slightly, if E is an AH-bundle we will often write ιE : E → † ∗ S † ∗ H ⊗ (E ) for the set of inclusion maps x∈M (ιEx : Ex → H ⊗ (Ex) ). We could broaden our definition of an AH-bundle, demanding only that E should be 0 0 a real vector bundle possessing a left H-action and a real subbundle E such that (Ex,Ex) is an AH-module for all x ∈ M. The following example shows why this definition would be unwieldy: Example 6.3.3 Let M = R, and define a real vector bundle E = R × H. Define a real subspace E0 ⊂ E by 0 Ex = hi1, i2 + xi3i. 0 It is easy to check that (Ex,Ex) = Xi3−xi2 as an AH-module. Thus E is not an AH- bundle, since the fibres Ex and Ey are not AH-isomorphic for x 6= y. There is good reason for excluding E from being an AH-bundle. For example, X if x = 0 (E⊗ X ) = i3 H i3 x 0 otherwise.

So despite the fact that E is a smooth vector bundle, E⊗HXq is not; if we considered this broader class of objects to be ‘AH-bundles’ then they would not be closed under the tensor product. k Let V be a real vector bundle over the manifold M with fibre R , and let P V be the bundle of frames associated with V . Then P V is a principal bundle with fibre V k GL (k, R), and V = P ×GL(k,R) R . In the same way, let E be an AH-bundle with fibre E W and let G = AutAH(W ). We can define a principal G-bundle P associated with E E so that E = P ×G W . As we know, G will be a subgroup of GL (k, H). In most cases, G will be significantly smaller than GL (k, H). For example, if W = U2j for some ∼ ∗ j then by Theorem 4.2.9 we have HomAH(U2j,U2j) = R, so AutAH(U2j) = R = R \{0} ∗ and P E is just a principal R -bundle. Principal bundles associated with AH-bundles tend to be much smaller than those associated with real or complex vector bundles.

6.3.1 Connections on AH-bundles

Deﬁnition 6.3.4 Let E be an AH-bundle over M. An AH-connection on E is an AH-morphism ∞ ∞ ∗ 1 ∇E : C (M,E) −→ C (M,E ⊗ T M) = Ω (M,E) which satisﬁes the rule ∇E(f · α) = df ⊗ α + f · ∇Eα for all α ∈ C∞(M,E), f ∈ C∞(M, H).

We will sometimes omit the subscript and just write ‘ ∇ ’ when the AH-bundle E † ∗ is clearly implied. Let ∇ be an AH-connection on E. Then the map id ⊗(∇E) is † ∗ an AH-connection on H ⊗ (E ) which preserves ιE(E). Thus every AH-connection ∇ on E is equivalent to a connection (∇†)∗ on the real vector bundle (E†)∗. We will not distinguish between ∇ and (∇†)∗, but will write ∇ for both. This equivalence allows us to construct AH-connections on tensor products, in very much the same way that Joyce constructs the AH-morphism φ⊗Hψ from AH-morphisms φ and ψ [J1, Deﬁnition 4.4].

91 Lemma 6.3.5 Let (E,M) and (F,N) be AH-bundles and let ∇E and ∇F be AH- connections on E and F . Then ∇ and ∇ induce an A -connection ∇ on the E F H E⊗HF AH-bundle (E⊗HF,M × N).

† ∗ † ∗ Proof. We regard ∇E as a connection on (E ) . The bundle ((E ) ,M) extends trivially to a bundle ((E†)∗,M × N) over M × N. Using the natural identification ∗ ∼ ∗ ∗ † ∗ T(m,n)(M × N) = TmM ⊕ Tn N, define a differential operator ∇E,M on ((E ) ,M × N), where ∇E,M differentiates only in the M-directions of M × N. In the same way define † ∗ a connection ∇F,N on ((F ) ,M × N) which differentiates only in the N-directions. Adding these gives a differential operator

∞ † ∗ † ∗ ∞ † ∗ † ∗ ∗ ∇E,M ⊗ id + id ⊗∇F,N : C ((E ) ⊗ (F ) ) −→ C ((E ) ⊗ (F ) ⊗ T (M × N)).

This operator is a connection on the real vector bundle (E†)∗ ⊗ (F †)∗, so the operator † ∗ † ∗ id ⊗(∇E,M ⊗ id + id ⊗∇F,N ) on the bundle (H ⊗ (E ) ⊗ (F ) ,M × N) is an AH- connection. Since ∇E is an AH-connection, the operator id ⊗∇E,M ⊗id maps sections of ιE(E)⊗ † ∗ † ∗ ∗ † ∗ (F ) to sections of ιE(E) ⊗ (F ) ⊗ T (M × N), and acts trivially on the (F ) factor; ∗ thus id ⊗∇E,M ⊗ id maps sections of E⊗HF to sections of E⊗HF ⊗ T (M × N). The same is true for id ⊗ id ⊗∇F,N , and so the AH-connection id ⊗(∇E,M ⊗ id + id ⊗∇F,N ) preserves the AH-subbundle (E⊗HF,M ×N), on which it is therefore an AH-connection. We call this connection ∇ . E⊗HF If E and F are AH-bundles over the same manifold M we can use Lemma 6.3.5 to define an AH-connection on the diagonal bundle (M,E⊗HF ). This process can be described in terms of principal bundles. 3 Suppose that φ ∈ × ∗ AutAH(W ) for some AH-module W . Then φ induces a real linear isomorphism (φ ) : † ∗ † ∗ (W ) → (W ) . In this way the action of G = AutAH(W ) on W gives rise to an action on (W †)∗. Let E be an AH-bundle with fibre W , and let P be the associated principal bundle with fibre G, so that E = P ×G W . An AH-connection ∇E on E gives rise to a connection DP on the principal bundle P , and vice versa. Since connections on principal bundles always exist, we can use this to guarantee that every AH-bundle has an AH-connection. Let F be another AH-bundle with fibre V , let H = AutAH(V ) and let Q be the associated principal bundle with fibre H, so that F = Q ×H V . An AH-connection ∇F on F gives rise to a connection DQ on the principal bundle Q. Consider the principal bundle P × Q, a bundle over M × N with fibre G × H. Since G acts on (W †)∗ and † ∗ H acts on (V ) , there is an induced action of G × H on W ⊗HV . The AH-bundle (E⊗HF,M × N) is the associated bundle (P × Q) ×G×H W ⊗HV . Since the connections DP and DQ define a unique connection DP ⊕ DQ on the product bundle P × Q, this gives rise to a connection on the associated bundle E⊗HF . Since G × H acts as AH- automorphisms on W ⊗HV , this induced connection is an AH-connection, which we can identify with the A -connection ∇ of Lemma 6.3.5. H E⊗HF 3A similar discussion for connections in complex vector bundles can be found in [K, §1.5].

92 6.4 Q-holomorphic AH-bundles

Using complex geometry as a model, we define a q-holomorphic AH-bundle over a hypercomplex manifold M to be one which is holomorphic with respect to each complex structure Q ∈ S2. An AH-bundle is q-holomorphic if and only if it admits a connection 2 whose curvature takes values only in E2,0 = Λ−. We define the q-holomorphic sections of a q-holomorphic vector bundle using a version of the Cauchy-Riemann-Fueter equations. Q-holomorphic sections have interesting algebraic properties, which generalise those of Joyce’s q-holomorphic functions. We give an interpretation of q-holomorphic sections in terms of quaternionic algebra and the q-holomorphic cotangent space, which makes the step-by-step correspondence with complex geometry extremely clear. Here is the main definition of this section: Definition 6.4.1 Let E be an AH-bundle over the hypercomplex manifold M, equipped with an AH-connection ∇. Suppose that ∇ gives E the structure of a holomorphic vector bundle over the complex manifold (M,Q) for all Q ∈ S2. Then (E, ∇) is a q-holomorphic AH-bundle. If we want to be really specific we can write (E, π, M, ∇) for our q-holomorphic bundle; on the other hand, if a connection is already specified, we can simply refer to the total space E as a q-holomorphic AH-bundle. The existence of such a connection ensures that the different holomorphic structures are compatible. A lot can be learned about q-holomorphic AH-bundles by studying the connection ∇.

6.4.1 Anti-self-dual connections and q-holomorphic AH-bundles Let E be a complex vector bundle over the complex manifold (M,I) and let ∇ be a connection on E. Just as we split the exterior diﬀerential d = ∂ + ∂, we can deﬁne E ∂E = π1,0◦∇ : C∞(E) → C∞(E⊗Λ1,0M) and ∂ = π0,1◦∇ : C∞(E) → C∞(E⊗Λ0,1M).

Proposition 6.4.2 The connection ∇ gives E the structure of a holomorphic vector E E bundle if and only if ∂ ◦ ∂ = 0, i.e. the (0, 2)-component of the curvature R of ∇ vanishes. Conversely, if E is a holomorphic vector bundle then E admits such a connection.

Proof. This is Propositions 3.5 and 3.7 of [K, p. 9]. Let E be a q-holomorphic AH-bundle over the hypercomplex manifold M, so E has the structure of a holomorphic vector bundle over (M,Q) for every complex structure Q ∈ S2. Let ∇ be a connection on E and let R be the curvature of ∇. Each 2 E E Q ∈ S deﬁnes a splitting ∇ = ∂Q + ∂Q, and so a decomposition of the curvature tensor 2,0 1,1 0,2 R = RQ + RQ + RQ . By Proposition 6.4.2, ∇ deﬁnes a holomorphic structure on 0,2 E with respect to Q if and only if RQ ≡ 0. If we reverse the sense of Q then we 0,2 2,0 reverse the decomposition of R, so R−Q = RQ . Thus ∇ gives E the structure of a 2 2,0 0,2 holomorphic vector bundle with respect to every Q ∈ S if and only if RQ = RQ = 0 for all Q ∈ S2, i.e. the curvature of ∇ is of type (1, 1) with respect to all Q ∈ S2.

93 A 2-form ω is of type (1, 1) with respect to every Q ∈ S2 if and only if it is 2 ∗ 2 ∗ annihilated by the action of sp(1) on Λ T M, so ω ∈ E2,0 ⊂ Λ T M. By analogy with the 4-dimensional theory (see Example 3.2.6), we call E2,2 the space of self-dual 2 2 2-forms Λ+ and E2,0 the space of anti-self-dual 2-forms Λ−. A connection ∇ whose ∞ curvature R takes values only in C (End(E)⊗E2,0) is therefore called an anti-self-dual connection. Several authors have considered self-dual and anti-self-dual connections, particulary on quaternionic K¨ahler manifolds: for example Galicki and Poon [GP], Nitta [N] and Mamone Capria and Salamon [MS]. This is partly because such connections give minima of a Yang-Mills funtional on M. It is important to note that Mamone Capria and Salamon refer to the bundle E2,0 as self-dual rather than anti-self-dual, and so refer to connections taking values in E2,0 as self-dual connections. There is no unanimous convention in the literature: our choice is made because we are using the conventions 0 j j 0123 that Ij(e ) = e not −e , and that the volume form e gives a positive orientation of H. We have established the following result:

Theorem 6.4.3 Let E be an AH-bundle over the hypercomplex manifold M equipped with an anti-self-dual AH-connection nabla. Then (E, ∇) is a q-holomorphic AH- bundle. Conversely, an AH-bundle E equipped with an AH-connection ∇ is q-holomorphic if and only if ∇ is anti-self-dual.

This is similar to the idea of hyperholomorphic bundles described by Verbitsky [V, §2]. Verbitsky considers the case where B is a Hermitian vector bundle over a hyperkähler manifold, so the fibres of such a bundle are not normally H-modules. There is no reason why his definition cannot be extended to hypercomplex manifolds, since as we have seen, the Hermitian inner product is not necessary to force the connection to be anti-self- dual if it is to be compatible with the 2-sphere of holomorphic structures. If E is a q-holomorphic AH-bundle then it is easy to see that H ⊗ (E†)∗ is also q-holomorphic, and (E†)∗ ⊗ C is a hyperholomorphic bundle. Conversely, if B is a hyperholomorphic bundle with a real structure σ such that B = Bσ ⊗ C, then H ⊗ Bσ is a q-holomorphic AH-bundle. Any AH-subbundle E of H ⊗ Bσ which is preserved by the anti-self-dual connection ∇ will also be q-holomorphic. The following Proposition gives further insight into the analogy between holomorphic and q-holomorphic bundles:

Proposition 6.4.4 Let (E, π, M, ∇) be a q-holomorphic AH-bundle over the hypercomplex manifold M. Then E is itself a hypercomplex manifold.

Proof. Consider the splitting of TE into horizontal and vertical subbundles TE ∼= E ⊕ TM, where the vertical subbundle E is naturally defined by the structure of E as an AH-bundle and the horizontal subbundle TM is defined by the connection. Define a hypercomplex structure (I1,I2,I3) on TE as follows. On the vertical subbundle isomorphic to E, the action of I1, I2 and I3 is given by the fixed left H-action of i1, i2 and i3 on the fibres. On the horizontal subbundle isomorphic to TM, define

94 the hypercomplex structure (I1,I2,I3) to be the horizontal lift of the hypercomplex structure on M. This deﬁnes an almost-hypercomplex structure on E. The integrability of this structure is guaranteed by the fact that (E,Ij) is a holomorphic vector bundle for j = 1, 2, 3.

Example 6.4.5 Let M be a hypercomplex manifold. Then the Obata connection ∇ on T ∗M is anti-self-dual and deﬁnes an anti-self-dual AH-connection on H ⊗ T ∗M, which is thus a q-holomorphic AH-bundle. Let A ⊂ H ⊗ T ∗M be the q-holomorphic cotangent space of M,

∗ A = {ω0 + i1 ⊗ ω1 + i2 ⊗ ω2 + i3 ⊗ ω3 : ω0 + I1ω1 + I2ω2 + I3ω3 = 0, ωj ∈ T M}.

∞ Let a = a0 +i1a1 +i2a2 +i3a3 ∈ C (M,A). Then ∇a0 +I1(∇a1)+I2(∇a2)+I3(∇a3) = 0 ∗ (where I1, I2 and I3 act on the A factor of A⊗T M ), since ∇Ij = 0 for j = 1, 2, 3. It follows that ∇a ∈ C∞(A⊗T ∗M). Hence the Obata connection deﬁnes an AH-connection on A. Since the Obata connection is anti-self-dual, A is a q-holomorphic AH-bundle.

Example 6.4.6 Let E and F be q-holomorphic AH-bundles. Then so are the various associated bundles E⊗ F , E ⊕ F ,Λk E and so on, all with their induced A - H H H connections.

6.4.2 Q-holomorphic sections

Q-holomorphic sections of AH-bundles are defined using a version of the Cauchy-Riemann- Fueter equations. We cannot automatically rewrite Equation (6.7) to define a Cauchy- Riemann-Fueter operator on a general AH-bundle E, because the fibres of E will not in general have a well-defined right H-action. Instead, we use the inclusion map ιE to manipulate sections of e in a more manageable form. Definition 6.4.7 Let E be an AH-bundle over the hypercomplex manifold M equipped ∞ with an AH-connection ∇. Let e ∈ C (M,E) so that ιE(e) = e0 + i1e1 + i2e2 + i3e3, ∞ † ∗ ej ∈ C (M, (E )) . Then e is a q-holomorphic section of E if and only if

∇e0 + I1(∇e1) + I2(∇e2) + I3(∇e3) = 0, (6.12)

∗ ∗ where I1, I2 and I3 act on the T M factor of E ⊗ T M. Let P(M,E) = P(E) be the space of q-holomorphic sections of E. Let P(E)0 = P(E) ∩ C∞(M,E0) so that P(E) is an AH-submodule of C∞(M,E). Q-holomorphic sections are the natural generalisation of q-holomorphic functions, which are precisely the q-holomorphic sections of the trivial bundle M × H equipped with the ﬂat connection. So PM = P(M × H). A general AH-bundle might have no q-holomorphic sections. However, if the AH- bundle (E, ∇) is q-holomorphic then E is holomorphic with respect to the complex structure Q, and admits holomorphic sections. It is easy to adapt Lemma 6.1.3 to show that the holomorphic sections of this holomorphic vector bundle give rise to q- holomorphic sections of E.

95 Not all q-holomorphic AH-bundles have interesting q-holomorphic sections. For example, the q-antiholomorphic cotangent space B ⊂ H ⊗ T ∗M is closed under the action of the Obata connection, and so is technically a q-holomorphic AH-bundle. Recall that

∗ B = {ω − i1 ⊗ I1(ω) − i2 ⊗ I2(ω) − i3 ⊗ I3(ω): ω ∈ T M}.

∞ Let b = b0 − i1I1b0 − i2I2b0 − i3I3b0 ∈ C (M,B), so b is q-holomorphic if and only if

∇b0 − I1∇I1b0 − I2∇I2b0 − I3∇I3b0 = 0.

2 Since ∇Ij = 0 and Ij = −1, this equation is satisﬁed if and only if ∇b = 0, and the AH-bundle B admits no non-trivial q-holomorphic sections. In fact, this follows from the fact that B0 = 0, and such behaviour is predicted and described by the algebra of the quaternionic cotangent space.

Q-holomorphic sections and the quaternionic cotangent space

Let E be a q-holomorphic AH-bundle over the hypercomplex manifold M. We can describe the q-holomorphic sections of E using the q-holomorphic cotangent space, just as we did for q-holomorphic functions in the previous chapter. The inclusion map ιE and the AH-connection ∇ (regarded as a connection on (E†)∗ ) define an AH-morphism ∞ ∞ † ∗ ∗ † ∗ ∇ ◦ ιE : C (M,E) → C (M, H ⊗ (E ) ⊗ T M). After swapping the H and (E ) factors, we have a copy of the quaternionic cotangent space H ⊗ T ∗M, which we can split into the q-holomorphic and q-antiholomorphic spaces A and B. Thus we have a ∞ ∞ † ∗ map ∇ ◦ ιE : C (M,E) → C (M, (E ) ⊗ (A ⊕ B)). We can now use the projections A B π and π of the previous chapter to split the action of ∇ ◦ ιE into two operators. Definition 6.4.8 Let (E, ∇) be a q-holomorphic AH-bundle. We define the pair of operators δE : C∞(E) → C∞((E†)∗ ⊗ A) and δ¯E : C∞(E) → C∞((E†)∗ ⊗ B) by

E A δ = (id(E†)∗ ⊗π ) ◦ ∇ ◦ ιE and ¯E B δ = (id(E†)∗ ⊗π ) ◦ ∇ ◦ ιE.

Hence we regard E as a subspace of H ⊗ (E†)∗, and then the action of ∇ on E splits into a q-holomorphic part δE and a q-antiholomorphic part δ¯E so that in eﬀect ∇ = δE + δ¯E. This is an exact analogue of the complex case where a connection ∇ splits as ∇ = π1,0 ◦ ∇ + π0,1 ◦ ∇. Let V be a holomorphic vector bundle with a E connection ∇ compatible with the holomorphic structure, so that ∂ = ∂. A section v ∈ C∞(V ) is holomorphic if and only if π0,1 ◦ ∇(s) = 0. In just the same way, the operator e 7→ ∇e0 + I1(∇e1) + I2(∇e2) + I3(∇e3) of Deﬁnition 6.4.7 is precisely the real part of δ¯E, and a section e ∈ C∞(E) is q-holomorphic if and only if δ¯E(e) = 0. This analogy goes further. In complex geometry, a section v ∈ C∞(V ) is holomorphic ∞ 1,0 if and only if ∂v = 0, which means that ∇v ∈ C (V ⊗C Λ M). Here is the quaternionic analogue of this statement:

96 Proposition 6.4.9 Let (E, π, M, ∇) be a q-holomorphic AH-bundle, and let e ∈ C∞(E) be a q-holomorphic section. Then

∞ ∇ ◦ ιE(e) ∈ C (E⊗HA), where A is the q-holomorphic cotangent space of M.

¯E E ∞ † ∗ Proof. Let e be q-holomorphic, so δ (e) = 0 and ∇ ◦ ιE(e) = δ (e) ∈ C (H ⊗ (E ) ⊗ T ∗M). Using the identification (A†)∗ ∼= T ∗M we can regard δE(e) as a section of H ⊗ (E†)∗ ⊗ (A†)∗. † ∗ E ∞ † ∗ The induced connection ∇ on (E ) preserves ιE(E), so δ (e) ∈ C (ιE(E)⊗(A ) ). E ∞ † ∗ E Clearly δ (e) ∈ C ((E ) ⊗ ιA(A)), since δ is defined by projection to this subspace. ∞ Thus ∇ ◦ ιE(e) ∈ C (E⊗HA), by Definition 4.1.4. In complex geometry, a holomorphic section v of a holomorphic vector bundle V is one whose covariant derivative ∇v takes values in the complex tensor product of V with the holomorphic cotangent space. In hypercomplex geometry, a q-holomorphic section e of a q-holomorphic vector bundle E is one whose covariant derivative ∇e takes values in the quaternionic tensor product of E with the q-holomorphic cotangent space.

6.4.3 Sections of Tensor Products For real and complex vector bundles, there is a natural inclusion C∞(M,E)⊗C∞(N,F ) ,→ C∞(M ×N,E ⊗F ) given by (e⊗f)(m, n) = e(m)⊗f(n). The quaternionic analogue of this map is more delicate. Let E and F be AH-bundles. We want to define an AH- ∞ ∞ ∞ morphism φ : C (M,E)⊗HC (N,F ) → C (M × N,E⊗HF ). The obvious difficulty is that for sections e and f of E and F respectively, there will not in general be a section e⊗Hf of the AH-bundle (E⊗HF,M × N). Instead we use a generalisation of Joyce’s map φ : PM ⊗HPN → PM×N (Definition 6.1.6). † Consider first the fibres Em and Fn for m ∈ M and n ∈ N. Let α ∈ Em and † ∞ β ∈ Fn. Define an AH-morphism αm : C (M,E) → H by αm(e) = α(e(m)) for all ∞ ∞ † e ∈ C (M,E). Then αm ∈ C (M,E) . Similarly, define the ‘evaluation at n ∈ N ’, ∞ † βn(f) = β(f(n)). Then βn ∈ C (N,F ) . The operators αm and βm are generalisations ∞ † of Joyce’s θm ∈ C (M, H) , introduced in Lemma 6.1.1. With H-valued functions † ∼ (sections of M × H ), the evaluation map θm generates the whole of Hm = R. For more general AH-bundles, we need to consider the combination of a point m at which † to evaluate sections, and an AH-morphism α ∈ Em, since we can no longer assume that † the map α = id generates the whole of Em. † † † For AH-modules U and V , recall the linear map λUV : U ⊗ V → (U⊗HV ) of † † † Equation (6.2). Thus there are linear maps λEm,Fn : Em ⊗ Fn → (Em⊗HFn) and ∞ † ∞ † ∞ ∞ † ∞ ∞ λC (M,E),C (N,F ) : C (M,E) ⊗ C (N,F ) → (C (M,E)⊗HC (N,F )) . ∞ ∞ Definition 6.4.10 Let ∈ C (M,E)⊗HC (N,F ). Define φE,F ()(m, n) ∈ Em⊗HFn by the equation

∞ ∞ λEm,Fn (α ⊗ β) · φE,F ()(m, n) = λC (M,E),C (N,F )(αm ⊗ βn) ·

† † for all m ∈ M, α ∈ Em and for all n ∈ N, β ∈ Fn.

97 Thus φE,F () defines a section of the AH-bundle (E⊗HF,M × N). By the same arguments as in Definition 6.1.6, φE,F () is smooth because it is a finite sum of smooth sections, and so we have a linear map

∞ ∞ ∞ φE,F : C (M,E)⊗HC (M,F ) → C (M × N,E⊗HF ).

It is easy to see that φE,F is an injective AH-morphism. We shall call φE,F the section product map for AH-bundles. If e ∈ C∞(M,E) and f ∈ C∞(N,F ) satisfy the conditions of Lemma 4.1.7 (so if 2 ιE(e) and ιF (f) both take values in Cq for some q ∈ S ) then the section φE,F (e⊗ f) is ∼ H the complex product of the sections, so φ(e⊗Hf)(m, n) = e(m)⊗Hf(n) = e(m)⊗Cq f(n). The section product map is the natural tool for relating tensor products of sections with sections of tensor products and allows us to treat sections of more complicated AH- bundles using similar techniques to those used by Joyce for q-holomorphic functions. It is well known that sums and tensor products of holomorphic sections are themselves holomorphic. The same is true for sums of q-holomorphic sections, and there is an analogous description for q-holomorphic sections of quaternionic tensor products, using the section product map φE,F . This is a generalisation of the fact (Deﬁnition 6.1.6) that

φ : PM ⊗HPN → PM×N .

Theorem 6.4.11 Let (E, π1,M, ∇E) and (F, π2,N, ∇F ) be q-holomorphic AH-bundles, and let P(M,E) ⊂ C∞(M,E) and P(N,F ) ⊂ C∞(N,F ) be the AH-modules of their q-holomorphic sections. Then

φE,F : P(M,E)⊗HP(N,F ) −→ P(M × N,E⊗HF ), where E⊗ F is equipped with the connection ∇ , so the section product map takes H E⊗HF q-holomorphic sections to q-holomorphic sections.

Proof. Consider the anti-self-dual connection ∇ = id ⊗(∇ ⊗ id + id ⊗∇ ). E⊗HF E,M F,N E⊗ F E⊗ F ∗ ∗ ∗ Then ∇ = δ H + δ¯ H . Using the natural splitting T (M × N) ∼= T M ⊕ T N, E⊗HF ¯E⊗ F ¯E,M ¯F,N ¯E,M we see that δ H = δ ⊗id + id ⊗δ , where δ differentiates sections of ιE(E) in the M directions and then projects to the q-antiholomorphic cotangent space of M ×N, and similarly for δ¯F,N . ∞ Let ∈ P(E)⊗HP(F ), and let φE,F () ∈ C (M ×N,E⊗HF ), where φE,F is the sec- † ∗ tion product map of Definition 6.4.10. Since ∈ ιP(E)(P(E)) ⊗ (P(F ) ) , it follows that ¯E,M ¯F,N ¯E⊗ F (δ ⊗id)(φE,F ()) = 0. Similarly, (id ⊗δ )(φE,F ()) = 0. Hence δ H (φE,F ()) = 0, and φE,F () is q-holomorphic. Just as for q-holomorphic functions, when M = N we can restrict to the diagonal bundle (E⊗HF,M). The restriction map ρ clearly preserves q-holomorphic sections, and we obtain a natural product ρ ◦ φE,F : P(M,E)⊗HP(M,F ) → P(M,E⊗HF ). In particular, let E be a q-holomorphic AH-bundle over M, let P(E) be the AH-module of q-holomorphic sections of E and let PM = P(H) be the H-algebra of q-holomorphic functions on M. Then by Theorem 6.4.11, we have a natural AH-morphism ∼ ρ ◦ φH,E : PM ⊗HP(E) −→ P(H⊗HE) = P(E). (6.13) We can descibe such an algebraic situation by saying that P(E) is an H-algebra module over PM . Here is the definition of an H-algebra module:

98 Axiom M. (i) Q is an AH-module and (P, µP ) is an H-algebra.

(ii) There is an AH-morphism µQ : P⊗HQ → Q, called the module multiplication map.

(iii) The maps µP and µQ combine to give AH-morphisms µP ⊗H id and id ⊗HµQ : P⊗HP⊗HQ → P⊗HQ. Composing with µQ gives AH- morphisms µQ ◦ (µP ⊗H id) and µQ ◦ (id ⊗HµQ): P⊗HP⊗HQ → Q. Then µQ◦(µP ⊗H id) = µQ◦(id ⊗HµQ). This is associativity of module multiplication.

(iv) For u ∈ Q, 1⊗Hu ∈ P⊗HQ by Lemma 4.1.7. Then µQ(1⊗Hu) = u for all u ∈ Q. Thus 1 acts as an identity on Q.

Deﬁnition 6.4.12 Q is an H-algebra module if Q satisﬁes Axiom M.

The idea of an H-algebra module was suggested by Joyce, and the axioms follow a very similar pattern to those for an H-algebra (Definition 6.1.4). It is relatively easy to see that the q-holomorphic sections P(E) of a q-holomorphic AH-bundle form an H-algebra module over the H-algebra PM , the module multiplication map being the section product map φH,E of Equation (6.13). The proof of this statement is obtained by adapting Joyce’s proof that the q-holomorphic functions PM form an H-algebra [J1, Theorem 5.5]. One possible application of this idea is to study anti-self-dual connections (instantons) on H. Different connections will give rise to different H-algebra modules of q-holomorphic sections. This suggests that the theory of H-algebra modules over PH might lead to an algebraic description of instantons on H.

99 Chapter 7

Quaternion Valued Forms and Vector Fields

This final chapter uses the algebra and geometry developed so far to describe quaternion- valued tensors on hypercomplex manifolds. On a hypercomplex manifold we have global complex structures, which allows us to adapt the real-valued double complex of Chapter ∗ ∼ 3 to decompose quaternion-valued forms. The splitting H ⊗ T M = A ⊕ B of the previous chapter is the first example of this type of decomposition. More generally, for all k and r we obtain a splitting of H ⊗ Ek,r into two AH-bundles. This decomposition gives rise to a double complex of quaternion-valued forms on hypercomplex manifolds, which has some advantages over the real-valued double complex. Not only the top row but the top two rows of the quaternion-valued double complex are elliptic, and in four dimensions the whole complex is elliptic. The top row of the quaternion-valued double complex is particularly well-behaved, and can be constructed using quaternionic algebra. This allows us to define q-holomorphic k-forms, and to describe their algebraic structure using ideas from the previous chapter. A similar approach to quaternion-valued vector fields is also fruitful. Just as with the quaternionic cotangent space, there is a splitting of the quaternionic tangent space H ⊗ TM using which we define a hypercomplex version of the ‘(1, 0) vector fields’ on a complex manifold. These vector fields are closed under the Lie bracket (suitably adapted to the quaternionic situation). This encourages us to adapt the axioms for a Lie algebra to form a quaternionic version, in a similar way to that in which Joyce arrived at H- algebras. The Lie bracket on hypercomplex manifolds defines a natural operation on vector fields which satisfies these axioms. In recent times, Spindel et al. [SSTP] and Joyce [J3] demonstrated the existence of invariant hypercomplex structures on certain compact Lie groups and their homogeneous spaces. This discovery presents us with lots of examples of finite-dimensional quaternionic Lie algebras. We use this idea to calculate the quaternionic cohomology groups of the group U(2), and suggest how these methods may be extended to higher-dimensional hypercomplex Lie groups.

100 7.1 The Quaternion-valued Double Complex

In Chapter 3, we saw how the real-valued exterior forms ΛkT ∗M on a quaternionic manifold M are acted upon by the principal Sp(1)-bundle Q of local almost complex k ∗ structures on M. Recall that the subbundle of Λ T M consisting of Vr-type represen- n tations is denoted Ek,r = εk,rVr. Suppose in addition that M is hypercomplex, and so possesses global complex structures I1, I2 and I3 = I1I2 (in other words, Q is the trivial bundle M ×Sp(1) ). Instead of just real or complex forms (which we can think of as taking values in the trivial representation V0), consider forms taking values in some Sp(1)-representation W , i.e. sections of the bundle W ⊗ ΛkT ∗M. Since the Sp(1)-action on ΛkT ∗M is now deﬁned by global complex structures, we can take the diagonal action under which the subspace W ⊗ Ek,r splits according to the Clebsch-Gordon formula. The situation in which we are particularly interested is that of forms taking values L R in the quaternions H = V1 ⊗ V1 . As suggested in Section 3.4, we obtain splittings by coupling the right Sp(1)-action on H with the Sp(1)-action on ΛkT ∗M. In symbols, this takes the form

∼ L R n G ∼ n L RG RG H ⊗ Ek,r = V1 ⊗ V1 ⊗ εk,rVr = εk,rV1 ⊗ (Vr+1 ⊕ Vr−1 ). (7.1)

Proposition 7.1.1 Let M 4n be a hypercomplex manifold. The AH-bundle H ⊗ ΛkT ∗M decomposes as k+1 ! k ∗ ∼ L M n n RG H ⊗ Λ T M = V1 ⊗ (εk,r+1 + εk,r−1)Vr , r=0 where r ≡ k + 1 mod 2.

Proof. By Proposition 3.2.1, we have

k ! k ∗ ∼ L R M n G H ⊗ Λ T M = V1 ⊗ V1 ⊗ εk,rVr , r=0 where the ‘Sp (1)G-action’ is the action of Sp(1) on ΛkT ∗M induced by the hypercomplex structure. Taking the diagonal Sp(1)RG-action using the Clebsch-Gordon formula, this becomes k ! k ∗ ∼ L M n RG RG H ⊗ Λ T M = V1 ⊗ εk,r(Vr+1 ⊕ Vr−1 ) . (7.2) r=0

Collecting together the Vr representations yields the formula in the Proposition.

n n k ∗ Deﬁnition 7.1.2 Deﬁne Fk,r to be the subspace (εk,r+2 +εk,r)V1 ⊗Vr+1 ⊆ H⊗Λ T M.

The primed part of the space Fk,r is obtained using the theory of Chapter 5. As in Equation 5.10, Equation 7.1 is a decomposition of H ⊗ Ek,r into stable and antistable ∗ ∼ AH-modules. (This is a generalisation of the splitting H ⊗ T M = A ⊕ B.) Deﬁning 0 k ∗ † ∗ Fk,r = Fk,r ∩ (I ⊗ Λ T M), the space Fk,r is an AH-subbundle of H ⊗ (Fk,r) = H ⊗

101 (Ek,r+2 ⊕ Ek,r), and each (fibre of the) AH-bundle Fk,r is the direct sum of stable and antistable components. The splitting ∼ n L RG RG H ⊗ Ek,r = εk,rV1 ⊗ (Vr+1 ⊕ Vr−1 ) gives an H-module isomorphism  n n × εk,rUr ⊕ εk,rUr−2 for r even ∼  H ⊗ Ek,r = (7.3)  1 n 1 n × 2 εk,rUr ⊕ 2 εk,rUr−2 for r odd. ∗ ∼ As with the decomposition H ⊗ T M = A ⊕ B, these are not AH-isomorphisms because some of the primed part is lost in the splitting. We give names to these spaces as follows (where as usual a = 1 if n is even and a = 2 if n is odd): ↑ 1 n ↓ Definition 7.1.3 Define Fk,r to be the AH-bundle a εk,rUr ⊆ H ⊗ Ek,r. Define Fk,r 1 n × ↑ to be the AH-bundle a εk,r+2Ur ⊆ H ⊗ Ek,r+2. Thus Fk,r is the Ur-type subspace of k ∗ ↓ × k ∗ H ⊗ Λ T M and Fk,r is the Ur -type subspace of H ⊗ Λ T M, so that

↑ ↓ Fk,r = Fk,r ⊕ Fk,r.

↑ The q-holomorphic cotangent space A is F1,1 = F1,1 and the q-antiholomorphic cotan- ↓ gent space B is F1,−1 = F1,−1.

↑ † ∗ ↓ † ∗ ∼ ↑ With these deﬁnitions, we have (Fk,r ) = (Fk,r−2 ) = Ek,r, and H ⊗ Ek,r = Fk,r ⊕ ↓ ∗ ∼ Fk,r−2. Just as with the splitting H ⊗ T M = A ⊕ B, there is an injective AH-morphism

↑ ↓ Fk,r ⊕ Fk,r−2 ,→ H ⊗ Ek,r which is an H-linear isomorphism of the total spaces but is not injective on the primed parts. A short calculation shows that

↑ n ↓ n dim Fk,r = 2(r + 2)εk,r dim Fk,r = 2(r + 2)εk,r+2 ↑ 0 n and ↓ 0 n dim Fk,r = (r + 3)εk,r dim Fk,r = (r + 1)εk,r+2.

↑ ↓ Since we can consider the bundles Fk,r and Fk,r separately, it may appear strange to mix up the stable and antistable fibres in the single bundle Fk,r. However, it soon becomes clear that exterior differentiation does not necessarily map stable fibres to stable fibres or antistable fibres to antistable fibres, so in order to obtain a double complex it is necessary to amalgamate the stable and antistable contributions. The definition of the operators δ and δ¯ of Section 6.2 can now be generalised to cover the whole of H ⊗ Λ•T ∗M, giving rise to a double complex of quaternion-valued forms. k ∗ Definition 7.1.4 Let πk,r be the natural projection map πk,r : H ⊗ Λ T M → Fk,r. Define the differential operators δ, δ¯ :Ωk(M, H) → Ωk+1(M, H) by ∞ ∞ ¯ ∞ ∞ δ : C (Fk,r) → C (Fk+1,r+1) δ : C (Fk,r) → C (Fk+1,r−1) and ¯ δ = πk+1,r+1 ◦ d δ = πk+1,r−1 ◦ d .

102 ∞ ∞ Theorem 7.1.5 The exterior derivative d maps C (M,Fk,r) to C (M,Fk+1,r+1 ⊕ Fk+1,r−1), so d = δ + δ.¯ It follows that δ2 = δδ¯ + δδ¯ = δ¯2 = 0. k ∗ Lk+1 and the decomposition H ⊗ Λ T M = r=0 Fk,r gives rise to a double complex.

Proof. The proof works in exactly the same way as that of Theorem 3.2.3. Let ∇ be ∞ ∞ ∗ the Obata connection on M, so ∇ : C (M,Fk,r) → C (M,Fk,r ⊗ T M). Since ∗ n n Fk,r ⊗ T M = (εk,r+2 + εk,r)V1 ⊗ Vr+1 ⊗ 2nV1, ∼ it follows (from the Clebsch-Gordon splitting Vr+1 ⊗ 2nV1 = 2n(Vr+2 ⊕ Vr), followed by ∞ ∞ the antisymmetrisation d = ∧◦∇ ) that d : C (M,Fk,r) → C (M,Fk+1,r+1 ⊕Fk+1,r−1). The rest of the theorem follows automatically.

Figure 7.1: The Quaternion-Valued Double Complex

∞ C (M,F3,3) ...... etc. ¨* H δ ¨¨ HH ¨ H ¨¨ HHj ∞ C (M,F2,2) ¨¨* HH ¨¨* δ ¨ H δ ¨ ¨¨ ¯ HH ¨¨ ¨ δ Hj ¨ ∞ ∞ C (M,F1,1 = A) C (M,F3,1) ¨¨* H ¨¨* H δ ¨ HH δ ¨ HH ¨¨ ¯ HH ¨¨ ¯ HH ¨ δ Hj ¨ δ Hj ∞ ∞ C (M,F0,0 = H) C (M,F2,0) H ¨* H ¨* HH δ ¨¨ HH δ ¨¨ ¯ H ¨ ¯ H ¨ δ HHj ¨¨ δ HHj ¨¨ ∞ ∞ C (M,F1,−1 = B) C (M,F3,−1) ...... etc.

As already hinted, the operators δ and δ¯ do not preserve stable or antistable sub- ¯ spaces. For example, though we have δ : Fk,r → Fk+1,r−1, it is not the case that ¯ ↑ ↑ δ : Fk,r → Fk+1,r−1. This can be observed in the simplest of cases — a quaternion-valued ↑ ¯ function f is a section of F0,0 = F0,0, and unless f is q-holomorphic δf is a nonzero ↓ section of F1,−1. Much of the theory from the real-valued double complex of Chapter 3 can be adapted to describe the quaternion-valued version as well. For example, the operators δ and δ¯ can be expressed in a similar fashion to D and D, using the Casimir element technique of Lemma 3.2.7. The Casimir operator in question is that of the diagonal Lie algebra action given by the operators

I(ω) = I1(ω) − ωi1 J (ω) = I2(ω) − ωi2 K(ω) = I3(ω) − ωi3.

103 of Equation (6.5). This leads to the following adaptation of Lemma 3.2.7:

∞ Lemma 7.1.6 Let α ∈ C (Fk,r). Then

1 1 δα = − r + (I2 + J 2 + K2) dα 4 r + 2 and 1 1 δα¯ = (r + 4) + (I2 + J 2 + K2) dα. 4 r + 2

The results on ellipticity in Section 3.3 can also be adapted to the new situation. We can infer that the operator δ is elliptic except at the bottom spaces F2k−1,−1 and F2k,0. Again, the operator δ is elliptic at some of these lowest-weight spaces for low exterior powers; in particular the leading edge of spaces Fk,k forms a complex which is elliptic throughout. Closer examination also reveals that the ‘second row’ of spaces Fk,k−2 is also an elliptic complex with respect to δ.

Lemma 7.1.7 The complex

∞ δ ∞ δ δ ∞ δ 0 −→ C (F1,−1) −→ C (F2,0) −→ ... −→ C (F2n+1,2n−1) −→ 0 is elliptic.

∞ Proof. Ellipticity at C (Fk,k−2) for all k ≥ 3 follows from a generalisation of the techniques used in the proof of Theorem 3.3.1. We need to show that the complex is ∞ ∞ ∞ elliptic at C (F1,−1) = C (B) and C (F2,0). As in Section 3.3, it is enough to choose some e0 ∈ T ∗M and show that the symbol sequence

σ σ σ 0 −→ F1,−1 −→ F2,0 −→ F3,1 −→ ... etc.

0 0 0 is exact, where σ(ω) = σδ(ω, e ) = πk+1,r+1(ωe ) for ω ∈ Fk,r. (As usual ωe means ω ∧ e0.) 0 Since δ = d on F1,−1 = B, we have σ(β) = βe for all β ∈ B. It follows from the expression for B in Equation (6.9) that σ : B → F2,0 is injective, so the complex is elliptic at B. 0 Let ω ∈ F2,0. Then σ(ω) = 0 if and only if ωe ∈ F3,−1, which is the case if and only if I(ωe0) = J (ωe0) = K(ωe0) = 0. The ﬁrst of these equations is

0 0 0 1 0 I1(ωe ) − ωe i1 = I1(ω)e + ωe − ωe i1 = 0.

It follows by taking exterior product with e0 that ωe10 = 0. The same arguments for J and K show that

σ(ω) = 0 =⇒ ωe10 = ωe20 = ωe30 = 0, and so ωe0 must be equal to zero and ω = γe0 for some γ ∈ H ⊗ T ∗M.

104 It remains to show that we can choose β ∈ B such that ω = βe0. Suppose instead 0 0 that ω = αe , with α ∈ A. Since the complex is elliptic at A it follows that αe ∈ F2,0 1 0 1 2 3 if and only if α = σ(q) = 4 q(3e + e i1 + e i2 + e i3) for some q ∈ H. But then 0 0 1 0 1 2 3 αe = βe , where β = 4 q(−e + e i1 + e i2 + e i3) ∈ B. Hence we can ﬁnd β ∈ B such that σ(ω) = 0 implies that ω = σ(β) for all ω ∈ F2,0, so the complex is exact at F2,0. This completes the proof. This is an unexpected bonus — on hypercomplex manifolds, the double complex of quaternion-valued forms has not just one but two rows which with respect to the operator δ are elliptic throughout, namely the top row Fk,k and the second row Fk,k−2. Example 7.1.8 Quaternion-valued forms in four dimensions

Figure 7.2: The Quaternion-Valued Double Complex in Four Dimensions

Λ2 A ∼= Y H ¨¨* HH δ ¨ H ¨¨ ¯ HH ¨ δ Hj ∼ A ∼= Z F3,1 = Z ¨* H ¨* H δ ¨¨ HH δ ¨¨ HH ¨ ¯ H ¨ ¯ H ¨¨ δ HHj ¨¨ δ HHj ∼ × ∼ F0,0 = H F2,0 = 3H ⊕ H F4,0 = H H ¨* H ¨* HH δ ¨¨ HH δ ¨¨ ¯ H ¨ ¯ H ¨ δ HHj ¨¨ δ HHj ¨¨ ∼ × ∼ × B = U−1 F3,−1 = U−1

In four dimensions the situation is particularly friendly towards quaternion-valued forms, because the whole double complex is elliptic. To show this, choose a standard 0 3 ∗ 0 j basis of 1-forms {e , . . . , e } for T M so that as usual Ij(e ) = e . Explicitly, we have ∗ ∼ H ⊗ T M = F1,1 ⊕ F1,−1, where 0 1 2 3 F1,1 = A = {q0e + q1e + q2e + q3e : q0 + q1i1 + q2i2 + q3i3 = 0}, 0 1 2 3 F1,−1 = B = {q0e + q1e + q2e + q3e : q0 = q1i1 = q2i2 = q3i3}. 2 ∗ ∼ Next, H ⊗ Λ T M = F2,2 ⊕ F2,0, where F = Λ2 A = {q e01+23 + q e02+31 + q e03+12 : q i + q i + q i = 0}, 2,2 H 1 2 3 1 1 2 2 3 3

↑ ↓ F2,0 = F2,0 ⊕ F2,0 01−23 02−31 03−12 01+23 02+31 03+12 = he , e , e iH ⊕ {q1e + q2e + q3e : q1i1 = q2i2 = q3i3}. 3 ∗ ∼ Lastly, H ⊗ Λ T M = F3,1 ⊕ F3,−1, where 123 032 013 021 F3,1 = {q0e + q1e + q2e + q3e : q0 + q1i1 + q2i2 + q3i3 = 0}, 123 032 013 021 F3,−1 = {q0e + q1e + q2e + q3e : q0 = q1i1 = q2i2 = q3i3}.

It follows that the symbol map σδ is an isomorphism between F3,−1 and F4,0, and so in four dimensions the entire quaternion-valued double complex is elliptic.

105 k 7.1.1 The Top Row ΛHA and Q-holomorphic k-forms In contrast with the the role of the Cauchy-Riemann operator ∂ in the Dolbeault complex, the Cauchy-Riemann-Fueter operator δ¯ in the quaternion-valued double complex (Figure 7.1) does not begin a longer elliptic complex. 1 On the other hand, the operator δ on functions does extend to give the elliptic complex

∞ δ ∞ δ δ ∞ δ 0 −→ C (F0,0) −→ C (F1,1) −→ ... −→ C (F2n,2n) −→ 0.

This section discusses these top row spaces Fk,k which are of particular interest. In complex geometry the Hodge decomposition of forms results immediately from the ∗ ∼ 1,0 0,1 k isomorphism C⊗T M = Λ ⊕Λ and the isomorphism in exterior algebra Λ (U⊕V ) = L p q p+q=k Λ U ⊗ Λ V . With hypercomplex geometry we are not quite so lucky, since the ∼ ∗ splitting A ⊕ B = H ⊗ T M is not an AH-isomorphism but rather an injective AH- morphism which is not surjective on the primed parts. Despite the fact that we can identify ⊗ ΛkT ∗M with Λk ( ⊗ T ∗M), the induced A -morphism H H H H ι :Λk (A ⊕ B) ,→ ⊗ ΛkT ∗M k H H is therefore not an isomorphism for k > 1. This is why we resort to our more complicated analysis of the Sp(1)-representation on H ⊗ ΛkT ∗M to discover the quaternionic version of the Dolbeault complex. In spite of this, the inclusion map ιk is still of special interest, because it describes explicitly the top row of the double complex.

Proposition 7.1.9 Let A be the q-holomorphic cotangent space of a hypercomplex manifold M. The inclusion map ι :Λk (A⊕B) ,→ ⊗T ∗M identiﬁes Λk A with the highest k H H H k ∗ space Fk,k ⊆ H ⊗ Λ T M.

Proof. There is a natural identiﬁcation Λk (A ⊕ B) = L (Λp A⊗ Λq B), and so H p+q=k H H H M ι : Λp A⊗ Λq B,→ ⊗ ΛkT ∗M. k H H H H p+q=k

Since B0 = {0}, B⊗ B = {0} and so Λk B = {0} for k > 1 ; also A⊗ B = {0}. Thus H H H Λk (A ⊕ B) = Λk A for k > 1, and the map ι :Λk (A ⊕ B) → ⊗ ΛkT ∗M is none H H k H H other than the normal inclusion map ι :Λk A → ⊗ (Λk A†)∗. It follows that there is H H H an A -submodule of ⊗ ΛkT ∗M which is isomorphic to Λk A. H H H ∼ k Nk ∼ Now, A = nU1, so by Theorem 5.2.1, Λ A ⊆ A = mUk for some m. (Recall H H that Uk = aV1 ⊗ Vn+1 where a = 1 for k even and a = 2 for k odd.) By Proposition 4.1.16, dim Λk (nU ) = 2(k + 2)2n. It follows that H 1 k 12n 2n Λk A ∼= U = V ⊗ V . (7.4) H a k k k 1 k+1

Thus ι (Λk A) must be a subspace of ⊗ ΛkT ∗M of this form. k H H 1In 1991, Baston [Bas] succeeded in extending the Cauchy-Riemann-Fueter operator to a locally exact complex with a diﬀerent construction involving second order operators.

106 2n From Deﬁnition 7.1.2, we see that Fk,k = k V1 ⊗ Vk+1 as well. Since there are no other representations of this weight in H ⊗ ΛkT ∗M, it follows that ι (Λk A) = F , k H k,k so there is a natural isomorphism ι :Λk A ∼= F . k H k,k Just as we deﬁne the q-holomorphic cotangent space A as being a particular submodule of H ⊗ T ∗M, so that (A†)∗ = T ∗M, we identify its exterior powers with the k ∗ corresponding AH-submodules of H ⊗ Λ T M. Thus we will omit to write the ‘ ιk ’, writing Λk A = F and (Λk A†)∗ = E . H k,k H k,k In Section 6.2.1 we showed that the q-holomorphic cotangent space A is generated over H by the various holomorphic cotangent spaces (Corollary 6.2.7). We generalise this result to higher exterior powers as follows:

Theorem 7.1.10 2 Let A be the q-holomorphic cotangent space of a hypercomplex manifold M 4n. Then (id ⊗ χ )(Λk A⊗ X ) = · ι (Λk,0). H q H H q H q Q It follows that X Λk A = · ι (Λk,0), H H q Q Q∈S2 i.e. the space Λk A is generated over by the spaces of (k, 0)-forms Λk,0. H H Q

Proof. Since Λk A ∼= 2nV ⊗ V , it follows that Λk A⊗ X ∼= 2nX . This is A - H k 1 k+1 H H q k q H isomorphic to the submodule (id ⊗ χ )(Λk A⊗ X ) ⊂ Λk A, which by Theorem 5.3.2 is H q H H q H generated over H by the weight-spaces of Q with extremal weight. Consider the weights of the action of Q ∈ S2 ⊂ sp(1). The highest-weight vectors in k ∗ k,0 k ∗ C ⊗ Λ T M are the (k, 0)-forms ΛQ , which are mapped to Cq ⊗ Λ T M by the map ι . Thus · ι (Λk,0) ⊂ (id ⊗ χ )(Λk A⊗ X ), and since dim Λk,0 = 2n we see that q H q Q H q H H q C Q k · ι (Λk,0) = (id ⊗ χ )(Λk A⊗ X ), H q Q H q H H q proving the first part of the Theorem. Since Λk A is stable, it is generated by these subspaces. The result follows. H Just as the q-holomorphic cotangent space A is our quaternionic analogue of the holomorphic cotangent space T ∗ M, the A -bundle Λk A is the quaternionic analogue 1,0 H H of the bundle of (k, 0)-forms Λk,0 = Λk T ∗ M. Both are formed in the same way using C 1,0 exterior algebra over their respective fields, and both form the ‘top row’ of their double complexes. This suggests a natural definition of a ‘q-holomorphic k-form’: Definition 7.1.11 Let M be a hypercomplex manifold. A quaternion-valued k-form ω ∈ Ωk(M, ) is q-holomorphic if and only if ω ∈ C∞(M, Λk A) and δω¯ = 0. The H H k AH-module of q-holomorphic k-forms on M will be written PM , so the H-algebra of 0 q-holomorphic functions on M is PM = PM . 2This theorem is really a quaternionic version of (Salamon’s) Equation 2.10, which is essentially the same result for complexified k-forms.

107 This gives rise to what we may call the q-holomorphic de Rham complex

d=δ 1 d=δ 2 d=δ d=δ 2n−1 d=δ 2n d=δ 0 −→ PM −→ PM −→ PM −→ ... −→ PM −→ PM −→ 0. (7.5) Just as (on a complex manifold) Λk,0 is a holomorphic vector bundle, it is easy to see that (Λk A, ∇) is a q-holomorphic A -bundle where ∇ is (the connection induced H H k by) the Obata connection on M. The q-holomorphic k-forms PM are precisely the q- holomorphic sections P(Λk A) of the A -bundle (Λk A, ∇) as introduced in Deﬁnition H H H 6.4.7, as the following Proposition demonstrates: Proposition 7.1.12 A quaternion-valued k-form ω ∈ C∞(M, Λk A) satisﬁes the H equation δω¯ = 0 if and only if ∇ω ∈ C∞(Λk A⊗ A), i.e. ω is a q-holomorphic section H H of (Λk A, ∇). H Proof. The ‘if’ part is automatic, since ∇ω ∈ C∞(Λk A⊗ A) implies that dω = ∧◦∇ω ∈ H H C∞(Λk+1A) and so δω¯ = 0. H The reverse implication is nontrivial and depends upon analysing the Sp(1)- representation on Λk A ⊗ T ∗M. Using Equation (7.4), we have H 2n Λk A ⊗ T ∗M ∼= V L ⊗ V M ⊗ 2nV G H k 1 k+1 1 2n ∼= 2n V L ⊗ (V MG ⊕ V MG). k 1 k+2 k The bundle Λk A⊗ A is precisely the higher-weight subspace 2n2nV L ⊗ V GH . The H H k 1 k+2 2n L GH complementary subspace 2n k V1 ⊗ Vk is revealed by Proposition 7.1.1 to be none k+1 ∗ other than the bundle Fk+1,k−1 ⊂ H ⊗ Λ T M. The component of ∇ω taking values ¯ in Fk+1,k−1 is of course δω, the vanishing of which therefore guarantees that ∇ω ∈ C∞(Λk A⊗ A). H H The q-holomorphic de Rham complex therefore inherits a a rich and interesting alge- k k+1 braic structure. We have already noted that d : PM → PM . It follows from the theory of q-holomorphic sections that q-holomorphic forms are closed under the tensor product. Explicitly, let

∞ k ∞ l ∞ k l φΛk A,Λl A : C (M, Λ A)⊗ C (M, Λ A) −→ C (M × M, Λ A⊗ Λ A) H H H H H H H H be the section product map (Deﬁnition 6.4.10). Deﬁne φk,l to be the restriction to C∞(M, Λk+lA), brought about by the restriction ρ : M × M → M to the diagonal H submanifold M ⊂ M × M followed by the skewing map ∧ : C∞(M, Λk A⊗ Λl A) → diag H H H C∞(M, Λk+lA). It follows (from Theorem 6.4.11 and Proposition 7.1.12) that H k l k+l φk,l : PM ⊗HPM −→ PM .

Thus the H-algebra structure on the q-holomorphic functions PM extends to one • • on the q-holomorphic k-forms PM , and we say that PM forms a diﬀerential graded H-algebra. It is well known that on a real or complex manifold M one can use exterior products over R or C to give an algebraic structure to de Rham or Dolbeault cohomology, which is often called the cohomology algebra of M. It may be that the diﬀerential graded • H-algebra structure on PM can be used to give a similar description of the quaternionic cohomology of a hypercomplex manifold.

108 7.2 Vector Fields and Quaternionic Lie Algebras

Quaternionic algebra can also be used to describe vector fields on a hypercomplex manifold M. Using similar ideas to those of Section 6.2, we define a splitting of the quater- ∼ nionic tangent space H ⊗ TM = Ab ⊕ Bb. Quaternion-valued vector fields which take values in the subspace Ab ⊂ H ⊗ TM turn out to be a good hypercomplex analogue of the ‘ (1, 0) vector fields’ in complex geometry. In particular, an almost hypercomplex structure is integrable if and only if these vector fields are closed under a quaternionic version of the Lie bracket operator. This encourages us to treat these vector fields as a quaternionic Lie algebra, a new concept which we proceed to define and explore.

7.2.1 Vector Fields on Hypercomplex Manifolds As so often, we take our inspiration from complex geometry. An almost complex structure ∼ 1,0 0,1 1,0 I on a manifold M defines a splitting C ⊗ TM = T M ⊕ T M, where T M is the holomorphic and T 0,1M the antiholomorphic tangent space of M. Let V be the set of smooth vector fields on M. The Lie bracket is a bilinear map from V × V to V. Let V1,0 = C∞(M,T 1,0M) be the vector fields of type (1, 0) on M. The almost complex structure I is integrable if and only if the Lie bracket preserves (1, 0) vector fields, which is expressed by the inclusion

[V1,0, V1,0] ⊆ V1,0. (7.6)

The obstruction to this equation is the Nijenhuis tensor NI which measures the (0, 1) component of the Lie bracket of two (1, 0) vector fields. We present a similar theorem for quaternionic vector fields on hypercomplex manifolds. Let M 4n be a hypercomplex manifold. We define a splitting of the quaternionic ∗ ∼ tangent space H ⊗ TM, which is roughly dual to the splitting H ⊗ T M = A ⊕ B of Section 6.2. 4n ∼ Definition 7.2.1 Let M be a hypercomplex manifold so that TM = 2nV1 as an Sp(1)-representation. The quaternionic tangent space H ⊗ TM splits according to the equation ∼ L R G ∼ L RG RG H ⊗ TM = V1 ⊗ V1 ⊗ 2nV1 = 2nV1 ⊗ (V2 ⊗ V0 ). L RG L RG Define the AH-subbundles Ab = 2nV1 ⊗ V2 and Bb = 2nV1 ⊗ V0 , using the natural definitions Ab0 = Ab ∩ (I ⊗ TM) and Bb0 = Bb ∩ (I ⊗ TM) = {0}. Then Ab is the q-holomorphic tangent bundle and Bb is the q-antiholomorphic tangent bundle of M.

This definition is perfectly natural, though not quite ideal from the point of view of quaternionic algebra. We would like Ab and Bb to be dual to the cotangent spaces A ∼ × ∼ × and B. However, whilst there are H-linear bundle isomorphisms Ab = A and Bb = B , these spaces are not isomorphic as AH-bundles, nor is there any fruitful way to alter the definitions to make them so. The Lie bracket of vector fields is a bilinear map [ , ]: V × V → V, and so defines a natural linear map λ : V ⊗ V → V. As with H-algebras, we will find it much more meaningful to talk about linear maps on tensor products, and we can use this formulation to obtain a quaternionic analogue of Equation (7.6).

109 Deﬁnition 7.2.2 Let M be a hypercomplex manifold with q-holomorphic tangent space Ab ⊂ H ⊗ TM. Let V = C∞(M,TM) be the vector space of smooth real vector ﬁelds on M.

Define VH = H ⊗ V to be the AH-module of smooth quaternion-valued vector fields ∞ ∞ on M, so VH = C (M, H ⊗ TM). Define VA = C (M, Ab) to be the AH-submodule of H-valued vector fields on M taking values in Ab ⊂ H ⊗ TM. We will refer to elements of VA as A-type vector fields.

The relationship between these vector ﬁelds and the map λ : V ⊗ V → V is par- ∼ ticularly interesting. Using the canonical isomorphism (H ⊗ TM)⊗H(H ⊗ TM) = H ⊗ TM ⊗ TM, the Lie bracket deﬁnes an AH-morphism (which we shall also call λ )

λ : VH⊗HVH → VH, which is eﬀectively a Lie bracket operation on quaternionic vector ﬁelds. Here is the quaternionic version of Equation (7.6). The formulation and proof is similar in spirit to Theorem 6.4.11.

Theorem 7.2.3 Let M be a hypercomplex manifold with q-holomorphic tangent space Ab, and let VA denote the space of A-type vector ﬁelds on M. Then the Lie bracket λ on quaternionic vector ﬁelds preserves VA, so that

λ : VA⊗HVA → VA.

Proof. To begin with, we use the section product map φ (Definition 6.4.10) to regard A,b Ab ∞ ∞ elements of VA⊗HVA as sections in C (Ab⊗HAb) ⊂ C (H ⊗ TM ⊗ TM). P ∞ Let vj, wj ∈ V be vector fields such that qj ⊗vj ⊗wj ∈ C (Ab⊗ Ab), which means P P j PH that qj ⊗ vj, qj ⊗ wj ∈ VA. We want to find an expression for λ( qj ⊗ vj ⊗ wj) = P qj ⊗ [vj, wj]. The Obata connection ∇ is an AH-connection on Ab (and in fact is anti-self-dual, so P ∞ that Ab is a q-holomorphic AH-bundle). Thus ∇( qj ⊗ wj) is an element of C (Ab ⊗ ∗ ∗ T M). Contracting the T M-factor with vj ∈ TM, it follows that X qj ⊗ ∇vj wj ∈ VA, and similarly X qj ⊗ ∇wj vj ∈ VA.

Since ∇ is torsion-free, the Lie bracket [vj, wj] is given by the diﬀerence ∇vj wj −∇wj vj. It follows immediately that X X λ qj ⊗ vj ⊗ wj = qj ⊗ (∇vj wj − ∇wj vj) ∈ VA, proving the theorem.

110 The reason why the hypercomplex structure must be integrable to obtain this result is that the integrability of I, J, and K ensures that the connection ∇ with ∇I = ∇J = ∇K = 0 is torsion-free, and otherwise we would not have [v, w] = ∇vw − ∇wv. Another way to understand this result is in terms of the decomposition of tensors with respect to diﬀerent complex structures. Just as in Theorem 7.1.10, the bundle Ab is generated over H by the tensors of type (1, 0) with respect to the diﬀerent complex structures, and the bundle Ab⊗HAb is generated by the tensors of type (2, 0). In other words, we have

X 1,0 X 1,0 1,0 Ab = H · ιq(TQ M) and Ab⊗HAb = H · ιq(TQ M ⊗ TQ M). Q∈S2 Q∈S2 If every complex structure Q ∈ S2 is integrable, we have

1,0 1,0 1,0 [VQ , VQ ] ⊆ VQ

2 1,0 for all Q ∈ S , where VQ denotes the vector ﬁelds which are of type (1, 0) with respect to the complex structure Q. Since the ﬁbres of Ab are stable, it follows that the

Lie bracket must map sections of Ab⊗HAb to sections of Ab.

7.2.2 Quaternionic Lie Algebras

The result of the previous section encourages us to think of the vector fields VA as a quaternionic Lie algebra with respect to the Lie bracket map λ. We describe this idea as an abstract algebraic structure, and see how it fits in with some of Joyce’s other algebraic structures over the quaternions. As always, we do not talk about bilinear maps, but rather about linear maps on tensor products. A quaternionic Lie algebra will be an AH-module A together with an AH-morphism λ : A⊗HA → A, whose properties reflect those of a Lie bracket on a real or complex vector space: namely antisymmetry and the Jacobi identity. Here are the axioms for a quaternionic Lie algebra:

Axiom QL. (i) A is an AH-module and there is an AH-morphism λ = λA : A⊗HA → A called the Lie bracket. (ii) S2 A ⊂ ker λ. Thus λ is antisymmetric. H (iii) The composition λ ◦ (id ⊗Hλ) deﬁnes an AH-morphism from A⊗ A⊗ A to A such that Λ3 A ⊂ ker(λ ◦ (id ⊗ λ)). This is the H H H H Jacobi identity for λ.

Deﬁnition 7.2.4 The pair (A, λ) is a quaternionic Lie algebra if it satisﬁes Axiom QL.

We will often refer to A itself as a quaternionic Lie algebra when the map λ is understood. Axiom QL(iii) is probably the least familiar-looking of these axioms. By way of explanation, let (V, λ) be a real or complex Lie algebra. Then in terms of tensor products the Jacobi identity is λ ◦ (id ⊗λ)(x ⊗ y ⊗ z + y ⊗ z ⊗ x + z ⊗ x ⊗ y) = 0.

111 Since we can identify x⊗y−y⊗x with x∧y, we see that the Jacobi identity is equivalent to the condition that λ ◦ (id ⊗λ)(x ∧ y ∧ z) = 0, so that Λ3V ⊂ ker(λ ◦ (id ⊗λ)). Example 7.2.5 Let M be a hypercomplex manifold. The space of quaternion-valued vector fields VH is a quaternionic Lie algebra with respect to the Lie bracket operator λ : VH⊗HVH → VH. Axiom QL follows from the corresponding identities satisfied by the Lie bracket on real vector fields. Theorem 7.2.3 now shows that the A-type vector fields VA form a quaternionic Lie subalgebra of VH. This is the quaternionic version of the well-known fact that on a complex manifold, the (1, 0) vector fields form a complex Lie subalgebra of the complex vector fields. Joyce has already considered the notion of a Lie algebra over the quaternions, but in a different fashion. In [J1, §6], he writes down axioms called Axiom L and Axiom P, which define quaternionic analogues of Lie algebras and Poisson algebras: but instead of a Lie bracket λ : A⊗ A → A, Joyce’s quaternionic Lie bracket is a map ξ : A⊗ A → A⊗ Y , ∼ H H H where Y = U2 is the AH-module of Example 4.1.2. The reason for this is that Joyce’s main purpose is to describe the algebraic structure of q-holomorphic functions on hyperkählermanifolds. Since a hyperkähler manifold M has three independent symplectic forms, two functions f and g have three different Poisson brackets and the H-algebra PM of q-holomorphic functions on M has three ∼ 3 independent Poisson structures. Using the fact that I = R , Joyce describes the three † ∗ ∼ Poisson brackets using a single map ξ : PM ⊗HPM → PM ⊗ I. Recalling that (Y ) = † ∗ V2 = I, we have a map ξ : PM ⊗HPM → ιPM (PM ) ⊗ (Y ) which is in fact an AH- morphism whose image is contained in PM ⊗HY . This is why, for Joyce, quaternionic Lie algebras and Poisson algebras are defined by an AH-morphism ξ : A⊗HA → A⊗HY . We can relate these two algebraic ideas very simply by choosing an AH-morphism † ∼ η : Y → H. The space of such AH-morphisms is of course Y = V2. Then for every ∼ AH-module A there is a map id ⊗Hη : A⊗HY → A⊗HH = A. Suppose that ξ : A⊗HA → A⊗HY satisfies Joyce’s Axiom L. Define a Lie bracket λ : A⊗HA → A by setting λ = (id ⊗Hη) ◦ ξ. It follows immediately that the pair (A, λ) is a quaternionic Lie algebra in the sense of Axiom QL. To go in the opposite direction, suppose that (A, λ) is a quaternionic Lie algebra, and consider the AH-module A⊗HY . In order to define an HL-algebra we need to form a map from (A⊗HY )⊗H(A⊗HY ) to (A⊗HY )⊗HY . Let τ be the natural isomorphism τ : A⊗HY ⊗HA⊗HY → A⊗HA⊗HY ⊗HY which interchanges the second and third factors. Define an AH-morphism

ξ = (λ⊗H id ⊗H id) ◦ τ :(A⊗HY )⊗H(A⊗HY ) → (A⊗HY )⊗HY.

Then ξ satisﬁes Joyce’s Axiom L, and the pair (A⊗HY, ξ) forms an HL-algebra. Pre- cisely which HL-algebras may be obtained from quaternionic Lie algebras and vice versa using these constructions remains open to question.

7.3 Hypercomplex Lie groups

As a ﬁnal example, we consider hypercomplex structures on compact Lie groups, and show how these give rise to ﬁnite-dimensional quaternionic Lie algebras. Since the early

112 1950s, the mathematical world has been aware that every compact Lie group of even dimension is a homogeneous complex manifold. 3 This was first announced by Samelson, whose proof is an extension of Borel’s celebrated result that the quotient of a compact Lie group by its maximal torus is a homogeneous complex manifold. In 1988 and 1992 respectively, Spindel et al. [SSTP] and Joyce [J3] demonstrated independently that these results extend to hypercomplex geometry. Joyce’s approach also gives hypercomplex structures on more general homogeneous spaces. It relies on being able to decompose the Lie algebra of a compact Lie group as follows: Lemma 7.3.1 [J3, Lemma 4.1] Let G be a compact Lie group, with Lie algebra g. Then g can be decomposed as n n X X g = b ⊕ dk ⊕ fk, (7.7) k=1 k=1 P where b is abelian, dk is a subalgebra of g isomorphic to su(2), b+ k dk contains the Lie algebra of a maximal torus of G, and f1,..., fn are (possibly empty) vector subspaces of g, such that for each k = 1, 2, . . . , n, fk satisfies the following two conditions: (i) [dl, fk] = {0} whenever l < k, and (ii) fk is closed under the Lie bracket with dk, and the Lie bracket action of dk on 2 fk is isomorphic to the sum of m copies of the basic representation V1 of su(2) on C , for some integer m. By adding an additional p (with 0 ≤ p ≤ Max(3, rk G) ) copies of the abelian Lie algebra u(1) to b if necessary, this decomposition allows us to define a hypercomplex structure on pu(1) ⊕ g as follows. For a 1-dimensional subspace bk of pu(1) ⊕ b there ∼ is an isomorphism bk ⊕ dk = R ⊕ I = H. The action of dk on fk gives an isomorphism ∼ m ∼ l fk = H . This gives an isomorphism pu(1) ⊕ g = H , in other words a hypercomplex structure. Normally there are many choices to be made in such an isomorphism, which give rise to distinct hypercomplex structures. That such a hypercomplex structure on the vector space pu(1)⊕g defines an integrable hypercomplex structure on the manifold U(1)p × G follows from Samelson’s original work on homogeneous complex manifolds. This leads to the following result: Theorem 7.3.2 [J3, Theorem 4.2] Let G be a compact Lie group. Then there exists an integer p with 0 ≤ p ≤ Max(3, rk G) such that U(1)p × G admits a left-invariant homogeneous hypercomplex structure. There is a strong link between these hypercomplex structures and the quaternionic Lie algebras of the previous section. Suppose that g is any real Lie algebra. Then gH ≡ H ⊗ g is a quaternionic Lie algebra because Axiom QL is obviously satisfied. The quaternionic Lie algebra structure of g is much more interesting when g admits a hypercomplex structure as described above. In this case we define the subspace

gA = {v0 + i1v1 + i2v2 + i3v3 : v0 + I1v1 + I2v2 + I3v3 = 0} ⊂ gH which we shall call the space of A-type elements of gH. Lie groups with integrable left- invariant hypercomplex structures then give rise to interesting quaternionic Lie algebras. 3Note that not all ‘Lie groups possessing a complex structure’ are complex Lie groups, because their multiplication and inverse maps might not be holomorphic.

113 Corollary 7.3.3 Let G be a hypercomplex Lie group with hypercomplex structure

(I1,I2,I3), and let gA ⊂ gH be the subspace of A-type elements of gH. Then gA is a quaternionic Lie subalgebra of gH.

Proof. By Theorem 7.2.3, the set of A-type vector fields on G is closed under the Lie bracket operator λ. Since λ also preserves the left-invariant vector fields g, it preserves gA. In this way, Joyce’s hypercomplex structures on compact Lie groups give rise to many interesting finite-dimensional quaternionic Lie algebras. We can also begin to calculate the quaternionic cohomology groups of these manifolds. On a Lie group G the exterior differential d on G-invariant k-forms can be expressed as a formal differential d :Λkg∗ → Λk+1g∗. The map d is induced by the dual of the Lie bracket, which is a linear map λ : g ⊗ g → g. Since λ is antisymmetric it is effectively a map λ :Λ2g → g, so its dual is the map d = λ∗ : g∗ → Λ2g∗. In practice, this means that dω(u, v) = ω([u, v]) ω ∈ g∗ v, w ∈ g. The map d extends to a unique antiderivation on Λ•g∗ in the usual fashion. Questions about the cohomology of G as a real, complex or hypercomplex manifold can then be rephrased in terms of the cohomology of the complex (Λ•g∗, d). This is true at least for G-invariant forms, which accounts for all the de Rham cohomology since it is known that every de Rham cohomology class has a G-invariant representative. This is less obvious for Dolbeault and quaternionic cohomology groups, whose theory is possibly more subtle for this reason. Example 7.3.4 Let M = U(2) be the unitary group in two dimensions, i.e. the 2 subgroup of GL(2, C) preserving the standard hermitian metric on C . It is well-known that U(2) is isomorphic to U (1) ×Z2 SU(2), which is diffeomorphic to the Hopf surface S1 × S3. The Lie algebra u(2) appears naturally in the form of Equation (7.7), thanks to the decomposition u(2) = u(1) ⊕ su(2). Let u(1) = he0i and su(2) = he1, e2, e3i, so that

[e0, ej] = 0 and [ei, ej] = εijkek where i, j, k ⊂ {1, 2, 3}. Let {eα} be the dual basis for g∗. It follows that de0 = 0 and dei = ej ∧ ek, where {i, j, k} is an even permutation of {1, 2, 3}. Thus e0 and e123 both generate de Rham cohomology classes, and we have b0(M) = b1(M) = b3(M) = b4(M) = 1, b2(M) = 0. This complex is best described using the structure of u(2) as a representation τ of su(2) = he1, e2, e3i. This subgroup acts on u(2) via the adjoint representation, so that τv(w) = [v, w] for v ∈ su(2), w ∈ u(2). Since 0 0 1 2 3 [v, e ] = 0, e generates a copy of the trivial representation V0 and he , e , e i is just ∗ ∼ the adjoint representation of su(2), which is V2. Thus T M = V0 ⊕ V2, and this induces • ∗ a representation of su(2) on Λ T M by the usual Leibniz rule τv(w1 ∧ w2) = [v, w1] ∧ w2 + w1 ∧ [v, w2]. This gives the following decompositions: ∗ 0 1 2 3 ∼ T M = he i ⊕ he , e , e i = V0 ⊕ V2 2 ∗ 23 31 12 01 02 03 ∼ Λ T M = he , e , e i ⊕ he , e , e i = V2 ⊕ V2 3 ∗ 032 013 021 123 ∼ (7.8) Λ T M = he , e , e i ⊕ he i = V2 ⊕ V0 4 ∗ 0123 ∼ Λ T M = he i = V0.

114 The benefit of this approach is that the map d is su(2)-equivariant. Thus once we have shown that de1 = e23, it follows immediately that d gives an su(2)-equivariant isomorphism d : he1, e2, e3i → he23, e31, e12i. This allows us to read off cohomological information, including the quaternionic co- k,r k,r homology groups HD (M) and Hδ (M). To do this we compare the decomposition of Equation (7.8) with the decomposition of ΛkT ∗M induced by the hypercomplex structure. In other words, we have two representations of sp(1) on T ∗M. The first is the ∗ ∼ representation T M = 2V1 defined by the hypercomplex structure, and the second ∗ ∼ is the representation T M = V0 ⊕ V2 given by the adjoint action of the subalgebra su(2) ⊂ u(2). The quaternionic cohomology of M is defined using the first action, but the d operator is best described using the second. By collating the two we obtain a complete picture of the situation. 01+23 02+31 03+12 In four dimensions, E2,2 is the space of self-dual 2-forms he , e , e i, where ab+cd ab cd ∗ 01 02 03 e = e + e . The image under d of T M is he , e , e i, and the projection π2,2 is an su(2)-equivariant surjection onto E2,2. The sequence

0 −→ E0,0 −→ E1,1 −→ E2,2 −→ 0

1 3 is therefore exact at E2,2, and we obtain the standard self-dual cohomology of S × S ,

0,0 1,1 0 ∼ 2,2 HD (M) = R HD (M) = he i = R HD (M) = 0. (7.9) Moving to quaternion-valued forms, it is easy to show that non-zero δ-cohomology occurs only at F0,0, F1,1, F3,−1 and F4,0, the sequence 0 → B → F2,0 → F3,1 → 0 being exact. Explicitly, we have

0,0 1,1 0 1 2 3 ∼ × Hδ (M) = H Hδ (M) = h3e + i1e + i2e + i3e iH = U−1 4,4 0123 ∼ 3,−1 123 032 013 021 ∼ × Hδ (M) = he iH = H Hδ (M) = he − i1e − i2e − i3e iH = U−1. (7.10) It follows from the property of ellipticity that the cohomology sequences should be exact, as indeed they are. However, they are clearly not AH-exact, because they are not exact on the primed parts. The quaternionic algebra of such phenonomena might be interesting and merit closer study. It would be desirable to extend this work to higher-dimensional groups and homogeneous spaces. The group SU(3) provides an interesting case. A similar analysis to that above may provide the correct results, though because of the additional four dimensions this option is difficult and complicated. The principle would nonetheless be very much the same, and essentially involves comparing different sp(1)-actions, one defined by Joyce’s hypercomplex structure and the other arising from the adjoint representation of a particular subalgebra su(2) ⊂ su(3). For example, it is easy to demonstrate that 1,1 Hδ (SU(3)) 6= 0. Quaternionic cohomology is in this case certainly not a subset of de Rham cohomology, since b1(SU(3)) = 0. A logical precursor to developing the quaternionic cohomology theory of hypercomplex Lie groups would be to understand thoroughly the Dolbeault cohomology of homogeneous complex manifolds. Surprisingly, this theory seems to be lacking or at best

115 extremely obscure. The theory for complex groups and homogeneous spaces is docu- mented in Bott’s important paper [Bot, §2], but the link between theory and results in this paper is quite opaque and certainly contains some mistakes. Pittie’s paper [P] uses a simpler bicomplex than Bott’s, conjecturing that the cohomology of these complexes is the same. Certain hypercomplex nilmanifolds discussed by Dotti and Fino [DF] might also provide fruitful rseults. This would be an interesting area for future research.

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